The exams: ENGM2 Mon 27 Jul · EM2 Tue 28 Jul · AE2 Thu 30 Jul · PE2 Wed 5 Aug · EP2 Fri 7 Aug · MS2A Mon 10 Aug (all 09:30). Target: a solid 50% in each. The front is brutal (100 of the 155 hours land before 30 Jul, about 8h/day); the back half is comfortable (4–5h/day).

Phase 1, the crunch: 18–29 Jul (~8h/day)

• 1Every morning 18–29 Jul: the 2h AE2 session (list below), fresh brain, before anything else. Its value is the SPACING; missing sessions cannot be made up by cramming.
• 218–21 Jul, ~6h/day: ENGM2 (~26h). Fourier (ride the momentum), then Laplace (most formulaic), then first-order ODEs. Ruthlessly skip the MVC tail (Stokes/divergence proofs) and second-order ODE depth.
• 322–25 Jul, ~6h/day: EM2 (~26h). Vikram chapters + the eight recycled paper patterns ONLY: Gauss sphere/coax, B-fields of wires, induction/emf, capacitor energy, dielectric sphere, Maxwell statements. No derivation completionism.
• 426 Jul: final ENGM2 drill (papers, timed). 27 Jul after the exam: final EM2 drill. 28 Jul afternoon + 29 Jul: AE2 finale (mixed tutorial runs under time).

The 12 AE2 morning sessions (2h each)

• 1Op-amp golden rules + ideal properties (op-amp unit, first sections)
• 2Inverting amplifier: derivation, then gain practice until automatic
• 3Non-inverting amplifier + voltage follower/buffer
• 4Summing and difference amplifiers
• 5Decibels unit: landmarks table + log rules
• 6The no-calculator dB conversions + cascaded stages
• 7Gain-bandwidth product + frequency-response basics
• 8Active filters: first-order low-pass and high-pass
• 9Filter practice + response sketches
• 10Op-amp non-idealities (offset, slew) light pass + tutorial questions
• 11Mixed tutorial run, timed
• 12Mixed past-paper op-amp/dB questions, exam conditions

Deliberately absent: BJT/FET small-signal depth. That is >50% material and it is the sacrifice that makes this schedule possible.

Phase 2, the recovery: 31 Jul–9 Aug (4–5h/day)

• 531 Jul–4 Aug: PE2 (~15h) + start EP2 alongside. PE2 bundle: waveform RMS, rectification with smoothing, transformers, one converter, one thermal chain.
• 65 Aug afternoon + 6 Aug: finish EP2 (~15h total). Bundle: number systems, IEEE-754 encode/decode, ALU traces, memory dumps, reading GPIO C. Pure pattern drilling.
• 77 Aug afternoon + 8–9 Aug: MS2A (~25h). Axial/torsion/bending stress + Mohr’s circle from solved examples until mechanical.

The three rules that decide it

• !The next ten days are FULL-TIME. One lost day before 27 Jul has nowhere to be absorbed.
• !EM2 must not leak backwards into ENGM2’s four days; the maths feeds everything and goes first.
• !The daily AE2 block happens even on bad days. Spacing is the whole strategy for that subject.

The task, from the general recipe (with $\omega =1$ and the integral already shrunk to where the wave is 1):

Step 1: the antiderivative, checked, not assumed

We need a function whose derivative is $\sin (nt)$. The candidate is $-\cos (nt)/n$. Check by differentiating it:

Chain rule pushes out a factor $n$ and a minus; they cancel against the built-in $-1/n$, leaving exactly $\sin (nt)$. So both the minus and the divide-by-$n$ are compulsory.

Step 2: substitute the limits

The bracket notation means “the antiderivative at the TOP limit, minus it at the BOTTOM limit”. Top is $t=\pi$, bottom is $t=0$:

Step 3: subtracting a negative becomes adding

The second bracket was minus a negative, so it flips to plus. This is where most sign errors are born.

Step 4: cos(0) = 1

Zero angle means no turn at all, and cosine of no turn is 1 (the very top of the trig-values table):

Step 5: combine over the shared denominator

Both terms sit over $n$, so they merge into one fraction, written with the positive term first:

Step 6: bring back the prefactor

The $1/\pi$ has been waiting outside the bracket the whole time. Multiplying two fractions multiplies their denominators:

Step 7: split by even and odd n

Now use $\cos (n\pi )={(-1)}^n$ (its own entry below): the numerator is $1-1=0$ for even $n$, and $1-(-1)=2$ for odd $n$:

Concrete values, to make the pattern real:

• 1$b_1=\frac{2}{\pi }$  (odd: numerator 2)
• 2$b_2=0$  (even: numerator 0, the harmonic is simply absent)
• 3$b_3=\frac{2}{3\pi }$  (odd again, and smaller: higher harmonics contribute less)

That is every step between the integral and the answer. The same seven-step shape runs every bₙ and aₙ in this course; only the wave’s constants and limits change.

First, your real question: is there a general formula?

Yes. Every “find the Fourier series” question, this one, Q2(b), the exam’s, is solved by the SAME four lines. Nothing about them changes between questions; you only swap in your wave’s brace and period:

Read the template in words: any repeating wave = a constant + a mix of sines and cosines at frequencies 1×, 2×, 3×… the fundamental. The three integrals (the “Euler formulae”) measure HOW MUCH of each ingredient your particular wave contains. ${\int }_T$ means “integrate across any one full period”. The recipe order is always: find $T$ and $\omega =2\pi /T$ first, then the three integrals, then assemble.

What each symbol is, for a beginner

• ∑$\sum$ means “add up”: the terms for $n=1,2,3,\dots$ one after another, forever.
• n$n$ is the harmonic number: which ingredient we are on. $n=1$ wiggles once per period, $n=3$ three times, etc.
• a,b$a_n,b_n$ are plain NUMBERS (one per $n$): the amount of the $n$-th cosine and sine in the mix. Finding them IS the whole question.
• ∫An integral is an AREA machine: $\int fdt$ totals the area under the curve. Multiplying $f$ by $\cos$ or $\sin$ before integrating makes it measure the OVERLAP between your wave and that ingredient (the deep reason is orthogonality, covered in the Fourier unit; for solving questions you only need to run the machine).

Now run the recipe on Q2(a), every move shown

Ingredients: $h=1$ on $(0,\pi )$, $0$ on $(\pi ,2\pi )$. Window length $T=2\pi$, so $\omega =2\pi /2\pi =1$, which is why every trig argument below is just $nt$.

Coefficient 1: a₀ (four small moves)

• 1Prefactor: $2/T=2/2\pi =1/\pi$. Just arithmetic.
• 2Limits shrink from $(0,2\pi )$ to $(0,\pi )$ because $h=0$ on the second half, and integrating zero adds zero. The brace’s value (1) replaces $h$ on the part that remains.
• 3$\int 1dt$ from 0 to π = area of a rectangle, height 1, width π = $\pi$. No calculus needed once you see the rectangle.
• 4The series uses $a_0/2=1/2$, the wave’s average height (the DC term entry below).

Coefficient 2: aₙ (why every cosine dies)

Moves: same prefactor and limit-shrink as before. The antiderivative of $\cos (nt)$ is $\sin (nt)/n$, divide by $n$ because differentiating $\sin (nt)$ spits out a factor $n$ (chain rule) that must be cancelled. Evaluate top minus bottom: $\sin (n\pi )$ is the sine of a whole number of half-turns, ALWAYS 0 (the trig-values table entry), and $\sin (0)=0$. So every single cosine coefficient is zero, which the hidden-symmetry argument predicted without integrating.

Coefficient 3: bₙ (the one that survives)

Two sign traps: the antiderivative of $\sin$ carries a built-in MINUS, and evaluating the bracket top-minus-bottom turns that into $1-\cos (n\pi )$. Every micro-move of that evaluation is worked one line at a time in the entry above this one. Then the even/odd split (the $\cos (n\pi )={(-1)}^n$ entry): even $n$ gives $1-1=0$; odd $n$ gives $1-(-1)=2$:

Assemble: pour the numbers back into the template

The template wants $a_0/2$ plus all the surviving terms. $a_0/2=1/2$; every $a_n$ is 0 (no cosines); $b_n$ is 0 for even $n$ and $2/n\pi$ for odd $n$, so only $n=1,3,5,\dots$ appear:

Sanity-read the answer: the ½ is the average height (right, the wave spends half its time at 1). Only sines, because wave-minus-mean is odd. Only odd harmonics, coefficients shrinking like 1/n because the wave has jumps. Every feature of the answer was predictable from the graph, which is the real power move in the exam.

The takeaway

• 1ONE general recipe solves every such question: $\omega =2\pi /T$, then the three Euler integrals, then assemble into the template. Memorise those four lines.
• 2What changes per question is only the BOOKKEEPING: your wave’s brace sets the integral limits and the constants inside.
• 3The recurring micro-skills: antiderivatives of $\cos /\sin (nt)$ (divide by $n$, sine’s carries a minus), $\sin (n\pi )=0$, $\cos (n\pi )={(-1)}^n$. Those three facts finish almost every coefficient in this course.

For reading the point ITSELF: correct, the middle of a flat piece always has a clean, unambiguous height, as far from every jump as possible. But there are two ways “pick the middle” can still bite, and one of them happens in your own Q2(b).

Caveat 1: your point is safe, but its MIRROR PARTNER might not be

The symmetry test uses two points: $t$ and $-t$. Choosing $t$ mid-shelf guarantees nothing about where $-t$ lands. Concrete case, the Q2(b) wave: $h=-1$ on $(-2,0)$, $+1$ on $(0,1)$, period 3, jumps at …, −2, 0, 1, 3, …

• ✗Middle of the WIDE shelf: $t=-1$. Perfectly readable ($h=-1$). But its partner is $+1$, which is EXACTLY the seam where the wave jumps from +1 down to −1. Partner unreadable, test dead on arrival.
• ✓Middle of the NARROW shelf: $t=0.5$. Partner $-0.5$ sits comfortably inside the wide shelf. Both readable: $h(0.5)=+1$, $h(-0.5)=-1$. Test works.

So the complete habit is: middle of a shelf for your point, then CHECK where the partner lands. If it hits a seam, slide your test time a little (0.4 instead of 0.5, say) or pick the other shelf, any interior pair works.

Caveat 2: a middle point that PASSES proves nothing

Safety of reading is not the same as strength of conclusion. If your middle-point test FAILS, done, the symmetry is disproved forever. If it PASSES, you have learned almost nothing, symmetry must hold at EVERY point, and one agreeing pair cannot certify that. Q2(b) again: points near the origin pass the odd test (heights flip nicely), yet the wave is not odd, which a second pair (like $t=1.5$) exposes. When a test passes, either test more pairs, or argue from the whole shape (equal widths? never negative? mirror image by construction?).

The complete checklist

• 1Pick $t$ strictly inside a shelf (middle = maximum safety margin). Never $0$, never a jump.
• 2Locate $-t$ (use periodicity if needed) and confirm IT is inside a shelf too. If not, adjust $t$.
• 3A FAIL is final; a PASS demands more evidence (another pair, or a whole-shape argument).

Short answer: the brace IS the formula. There is no single algebraic expression like $h(t)=t^2$ hiding somewhere; the piecewise rule is the complete definition, and “plugging in” works differently for it.

Two kinds of function definitions

• 1Formula-functions, like $f(t)=t^2$: plugging in means COMPUTING. Feed in 3, do the arithmetic, get 9.
• 2Rule-functions (piecewise), like our $h$: plugging in means LOOKING UP. Feed in a time, check which case it falls into, read off that case’s value. No arithmetic on the value at all.

Both are complete, legitimate definitions. Maths does not require a function to be one algebraic expression; it only requires that every input gets exactly one output. The brace does that job by cases.

The FULL definition of h has two parts

The brace covers one window, $0<t<2\pi$. The periodicity statement on the right extends it to every other time: any input outside the window is first slid by whole periods until it lands inside. Together they define $h$ for all $t$.

The plug-in procedure, as a machine

The three plug-ins we actually did, run through the machine

• 1$h(\pi /2)$: already inside the window (1.57 is between 0 and 6.28). Which case? 1.57 < 3.14, first line. Output 1.
• 2$h(-\pi /2)$: outside the window (negative). Slide +2π: lands at $3\pi /2$ (4.71). Which case? Between 3.14 and 6.28, second line. Output 0.
• 3For practice, $h(7\pi /2)$ (≈ 11.0): outside (too big). Slide −2π: $3\pi /2$. Second line. Output 0.

So what did “put in our two numbers” mean in the even test?

The equation $h(-t)=h(t)$ is a CLAIM to be checked, not a formula to substitute into. Checking it at $t=\pi /2$ means: evaluate the left side with the lookup machine ($h(-\pi /2)=0$), evaluate the right side with the lookup machine ($h(\pi /2)=1$), and compare the two finished numbers: $0\ne 1$, claim false. Every “plug in” in the symmetry entries was this lookup, run once per side.

Perspective: plenty of engineering functions are lookup-defined rather than formula-defined, gear ratios per gear, tax bands, a thermostat’s on/off state. The square wave is just the simplest lookup function in your course, and the whole point of the Fourier series is to REBUILD it out of formula-functions (sines and cosines), because formulas are what calculus can differentiate, integrate and manipulate.

The middle is not compulsory, ANY time strictly inside a shelf works ($1$, $2$, $0.4$, all fine, see the “is the 2 arbitrary?” entry). The middle is just the safest spot. But $0$ and $\pi$ specifically are the two WORST choices, each broken for its own reason.

Reason 1: 0 and π are jump instants, there is no height to read there

Look at the brace: $h=1$ for $0<t<\pi$ and $h=0$ for $\pi <t<2\pi$. Both inequalities are STRICT ($<$, not $\le$), so neither line covers $t=0$ or $t=\pi$ at all. The function is literally undefined at the jumps. And you can see why on the graph: approach $\pi$ from the left and the wave sits at 1; approach from the right and it sits at 0. Two different answers collide at the same instant, so “the height at π” has no single value. A symmetry test needs definite heights to compare; a jump cannot supply one.

(Extra wrinkle: the Fourier series itself, evaluated at a jump, converges to the MIDPOINT ½, a third value different from both shelves. Jumps are genuinely ambiguous territory, stay off them.)

Reason 2: t = 0 is useless for a symmetry test even if it were defined

The test compares the height at $t$ with the height at $-t$. Put $t=0$ in and the mirror partner is $-0=0$: the point is its own partner. The even test becomes “is $h(0)=h(0)$?”, which is true for EVERY function ever. It can never expose an asymmetry, so it carries zero information. A useful test point must be strictly away from zero so that $t$ and $-t$ are genuinely different places.

Reason 3: π’s mirror partner is ALSO a jump

Suppose you shrugged off Reason 1 and tried $t=\pi$ anyway. The test then needs the height at $-\pi$, and by periodicity $-\pi$ sits exactly where $+\pi$ does in the repeating pattern: on ANOTHER seam between a 1-shelf and a 0-shelf. Both ends of the comparison are ambiguous. Nothing comparable on either side.

The rule to carry

• 1Pick test times STRICTLY INSIDE a flat stretch, never at a jump, never at $t=0$.
• 2Check where the MIRROR PARTNER lands too: it must also sit inside a shelf, not on a seam.
• 3“Middle of the shelf” is a habit, not a law: it keeps you (and your partner point) as far as possible from every edge, so there is never doubt about which shelf you are reading.

Companion to the graph-building entry below: same wave ($h=1$ on $(0,\pi )$, $0$ on $(\pi ,2\pi )$, repeating every $2\pi$), and now the question “is it odd, even, or neither?” answered with nothing assumed.

Step 0: what the two symmetries even are

“Even” and “odd” are names for two ways a graph can be its own twin:

• EEVEN = mirror twin. Fold the picture along the vertical axis (the line at $t=0$). If the left half lands exactly on the right half, the function is even. In symbols: the height at $-t$ always equals the height at $t$, written $h(-t)=h(t)$.
• OODD = upside-down twin. Spin the whole picture half a turn (180°) about the origin point. If it lands on itself, the function is odd. In symbols: the height at $-t$ is always the NEGATIVE of the height at $t$, written $h(-t)=-h(t)$.

A function might have the first symmetry, the second, or NEITHER. To find out which, you compare heights at mirror-partner times: some time $t$ on the right of zero, and its partner $-t$ the same distance to the left. One pair of heights is enough to kill a symmetry, because a symmetry must hold everywhere; one place where it fails means it is gone.

Step 1: pick a test time and read the right-side height

Pick $t=\frac{\pi }{2}$ (≈ 1.57), a convenient spot in the middle of the ON shelf. From the graph (or the brace: 1.57 is between 0 and 3.14, the “height 1” stretch):

Step 2: read the left-side height, slowly (this is the step people skip)

Now we need the height at the mirror partner, $t=-\frac{\pi }{2}$ (≈ −1.57). Problem: the brace only describes times between $0$ and $2\pi$, and −1.57 is outside that window. The way in is periodicity: the wave repeats every $2\pi$, meaning sliding any time forward by one full period lands on an identical part of the wave. So slide −π/2 forward by $2\pi$:

(The arithmetic: $-\frac{\pi }{2}+2\pi =-\frac{\pi }{2}+\frac{4\pi }{2}=\frac{3\pi }{2}$, about 4.71.) And 4.71 lies between $\pi$ (3.14) and $2\pi$ (6.28), the OFF shelf, height 0:

Step 3: run the EVEN test

Even demands mirror partners have EQUAL heights: $h(-t)=h(t)$. Put in our two numbers:

The left dot is on the floor, the right dot is up at 1. Fold the picture along the vertical axis and those two points do NOT land on each other. Not even.

Step 4: run the ODD test

Odd demands mirror partners have OPPOSITE heights: $h(-t)=-h(t)$. Since $h(\pi /2)=1$, oddness would need the partner to be at $-1$:

There is also a zero-effort way to see oddness was never possible: an odd graph must dip BELOW the axis somewhere (every up-height needs a matching down-height after the 180° spin). This wave lives entirely at heights 0 and 1, never negative, so it cannot be odd no matter which points you test. Not odd.

Step 5: the verdict, and what it changes

• 1Even test failed, odd test failed → the wave is NEITHER. That one word is what the exam wants stated (with a reason, e.g. the failing pair of heights).
• 2Consequence for the series: no symmetry shortcut applies directly, so in principle all three coefficient families ($a_0$, $a_n$, $b_n$) must be computed from the brace.
• 3Hidden bonus (worked in the DC-term entry below): subtract the mean ½ and what remains IS odd, which is the deep reason the $a_n$ all come out zero anyway and the final series is a constant plus sines.

One caution worth engraving: a symmetry test that PASSES at one point proves nothing (symmetry must hold at every point), but a test that FAILS at one point is final. That is why one carefully chosen pair of heights settles “not even” and “not odd” here, yet no number of passing points could ever prove a symmetry on its own.

They are the same quantity, not merely related. In mechanics, angular velocity is how fast something turns: radians of angle per second. The Fourier $\omega$ is the angular velocity of an invisible rotating point whose shadow draws the wave. Same symbol, same formula, same units, on purpose.

The mechanics version first

A wheel turning steadily sweeps $2\pi$ radians (one full turn) every revolution. If one revolution takes $T$ seconds, the turning rate is:

That IS the formula from the tutorial. Nothing about it is special to waves yet, it is just “how fast is the turning”.

The bridge: a sine wave is circular motion seen edge-on

Take a point moving round a circle of radius $A$ at a steady $\omega$ rad/s. After $t$ seconds it has turned through the angle $\omega t$. Now watch only its height (its shadow on a vertical wall): that shadow is exactly $A\sin (\omega t)$. Every sinusoid is a rotation viewed from the side:

One full lap is $2\pi$ radians of turning AND one full period $T$ of the wave, which is exactly why $\omega =2\pi /T$ for waves too. The wave inherits its $\omega$ from the circle that generates it.

So what does ω = 1 rad/s mean for the square wave?

• 1The square wave itself does not rotate. But its Fourier series is built from harmonics $\sin (nt)$, $\cos (nt)$, and EACH of those is the shadow of its own spinning point.
• 2$\omega =1$ rad/s says the fundamental’s circle turns 1 radian per second, so a full lap ($2\pi$ rad) takes $2\pi$ seconds, matching the wave’s period $T=2\pi$. Consistent.
• 3The $n$-th harmonic’s circle spins $n$ times faster ($n\omega$ rad/s), completing $n$ laps per wave period. That is what “harmonics at multiples of the fundamental” means physically.

Why engineers keep ω around (and not just ν in Hz)

Frequency $\nu =1/T$ counts full turns per second (Hz); $\omega =2\pi \nu$ counts radians per second. Maths prefers $\omega$ because the argument of sin/cos must be an angle, writing $\sin (\omega t)$ keeps the bookkeeping in radians with no stray $2\pi$ factors. And the rotation picture is not a metaphor you discard later: it is literally the phasor used for AC circuits in PE2/AE2 (UK mains: $\nu =50$ Hz means a phasor spinning at $\omega =2\pi ·50\approx 314$ rad/s), and the same $\omega$ is a motor shaft’s angular velocity in dynamics. One concept, whole degree.

Goal: start from a blank page and end at the app’s graph, with no step assumed. The function is the one from Q2(a):

Step 0: what a graph even is

A graph is a lookup table drawn as a picture. The horizontal axis is the floor: every position on it is a time $t$. The vertical axis is the wall: every position on it is a height. A point drawn at floor-position $t$ and wall-height $y$ is the statement “at time $t$, the function’s value is $y$”. Drawing the whole graph means doing that for every time at once.

Before drawing anything, mark the landmark times the brace mentions. $\pi$ is just the number 3.14…, so the landmarks are $0$, $\pi \approx 3.14$, and $2\pi \approx 6.28$:

Step 1: draw the first line of the brace

Line one says: for every time between $0$ and $\pi$, the height is $1$. Pick any such time and plot its point: at $t=1$, height 1. At $t=2$, height 1. At $t=0.5$, height 1. Every single point lands at the SAME height, so joining them gives a flat shelf at height 1 running from $t=0$ to $t=\pi$:

Step 2: draw the second line, height 0 lies ON the axis

Line two says: for every time between $\pi$ and $2\pi$, the height is $0$. Height zero is not “nothing drawn”, it is a real shelf that happens to sit at the level of the floor itself. So the graph between $\pi$ and $2\pi$ is a flat segment lying exactly ON the t-axis. This is why the app’s picture looks like the wave “disappears” there: it has not disappeared, it is at height 0.

At $t=\pi$ itself, the value jumps: just left of $\pi$ the height is 1, just right of it the height is 0. Nothing is drawn vertically at the jump, a function has exactly one value per time, so there is no wall connecting the shelves. The dashed vertical in the app is only a visual guide marking where the teleport happens. (Fine print: the brace uses strict < signs, so at exactly $t=\pi$ it does not define a value at all; for Fourier purposes the series will later converge to the midpoint ½ there.)

The window we have drawn runs from 0 to $2\pi$, length $2\pi$. That length is the period: $T=2\pi$, hence $\omega =\frac{2\pi }{T}=1$ rad/s.

Step 3: photocopy the block both ways forever

“Periodic” means $h(t+2\pi )=h(t)$: shift the picture by one period and it must look identical. So stamp the Step-2 block at every shift of $2\pi$. Shift right: shelf at 1 on $(2\pi ,3\pi )$, zero on $(3\pi ,4\pi )$. Shift left: shelf at 1 on $(-2\pi ,-\pi )$, zero on $(-\pi ,0)$:

Step 4: verify against the app’s picture

• 1Just right of 0: shelf at 1, width π. Matches (the app’s hump from 0 to π). ✓
• 2Just LEFT of 0: the app shows the wave on the axis from −π to 0. Our copy rule predicts exactly that: $(-\pi ,0)$ is the zero shelf of the left-shifted block. ✓
• 3Spot values: $h(\pi /2)=1$ (inside the first hump), $h(3\pi /2)=0$ (on the zero shelf), matching the earlier reading-a-value entry. ✓

Contrast with Q2(b) below: same recipe exactly (draw the brace once, tile by the period), only the ingredients differ, Q2(b)’s shelves are −1 and +1 with unequal widths and the stamp distance is 3 instead of 2π. Master the recipe once and every “sketch over a few periods” question is the same two moves.

Yes, almost all of them, because the series is computed from the BRACE (the piecewise definition), not from your drawing. But there is exactly one pipe connecting the two, and if you use it carelessly a bad graph can poison the algebra. Know where that pipe is.

What each part is actually marked on

• 1The sketch marks are their own little pot: axes, correct widths and heights, a few periods shown. Lose these and they are gone, but they are only the sketch’s pot.
• 2The series marks come from: correct $T$ and $\omega =2\pi /T$, correctly split integrals with the right limits and the right constant in each piece, correct integration, correct assembly. Every one of those reads straight off the brace: for Q2(b), $b_n=\frac{2}{3}[{\int }_{-2}^0(-1)\sin (n\omega t)dt+{\int }_0^1(+1)\sin (n\omega t)dt]$. No picture required anywhere in that.

The one way a bad graph poisons the series

The symmetry shortcut. If your (wrong) sketch convinces you the wave is odd, you will write “odd, so $a_0=a_n=0$” and skip two integrals. If the wave is actually “neither” (as in Q2(b)), you have now thrown away real, non-zero coefficients, and the final series is wrong for a reason that traces straight back to the graph.

The safe play when unsure of your sketch

• ✓Don’t claim a symmetry you can’t verify from the brace. The algebraic test (compare h(−t) with ±h(t) using the piecewise rule, not the picture) is sketch-proof.
• ✓If in any doubt, compute all three coefficients from the brace. Costs a few minutes, immune to drawing errors, and every correct integral earns its marks even if the sketch above it is nonsense.
• ✓Use the graph only for free CHECKS, never as an input: the DC term should match the eyeballed average, and coefficients should decay like 1/n for a jumpy wave. If they disagree, something is wrong somewhere, which is information, not marks lost.

Bottom line for Q2(b): a mangled sketch costs you the sketch’s own couple of marks and nothing else, PROVIDED your series work starts from the brace and you only state “neither odd nor even” if the algebra (or the widths in the brace: 1 vs 2) backs it.

The question hands you a piecewise rule for ONE period and expects you to turn it into a repeating graph. There are exactly two steps: (1) draw literally what the brace says, once; (2) photocopy that block left and right forever. Here is each step slowly.

Step 1: read the brace, draw one block

The rule is:

Read each line as “flat shelf at this height, over this stretch of floor”: a shelf at height $-1$ from $t=-2$ to $t=0$ (width 2), then a shelf at height $+1$ from $t=0$ to $t=1$ (width 1). Together they cover $-2<t<1$, a window of total length $3$. That length IS the period: $T=1-(-2)=3$, so $\omega =\frac{2\pi }{3}$ rad/s.

Step 2: photocopy the block every 3 units

“Periodic with period 3” means $h(t+3)=h(t)$: slide the whole picture 3 to the right (or left) and it must look identical. So take the block from Step 1 and stamp copies at every shift of 3:

Where each feature of the final graph comes from:

• 1Narrow tops, wide bottoms: straight from the brace, the $+1$ piece is width 1, the $-1$ piece is width 2. (A one-third duty cycle.)
• 2Jump locations: the block’s internal jump at $t=0$ and its seams at $-2$ and $1$, then all of these repeated every 3: …, −2, 0, 1, 3, 4, 6, 7, …
• 3Dashed verticals: not part of the function, just guides marking the instant the value teleports between −1 and +1. A function can only have one value per t, so the jump has no “vertical wall”.

Why the graph shows “neither odd nor even” at a glance

The widths give it away. Mirror the graph in the vertical axis: just right of 0 the wave is $+1$ for a width-1 stretch, but just left of 0 it is $-1$, so it is not even. For oddness, beware: near the origin it LOOKS odd (heights flip across 0). Test one point further out: $h(1.5)=-1$ (in the (1,3) low shelf) but $-h(-1.5)=-(-1)=+1$. Not equal, so not odd. The unequal widths (1 vs 2) are what break both symmetries.

Lesson inside the lesson: a symmetry test that PASSES at one point proves nothing, symmetry must hold everywhere, and this wave passes the odd test near 0 yet fails it at 1.5. One failing point disproves; one passing point does not prove.

Bonus: what the graph already tells you about the series

Mean value by areas: over one period, area $=(2\times (-1))+(1\times (+1))=-1$, divided by the width 3 gives the DC term $-\frac{1}{3}$. The wave spends more time low than high, so its average sits below zero, and with no symmetry, expect both $a_n$ and $b_n$ to survive in the series.

This is from the both-odd-and-even proof below: we reached $f(t)=-f(t)$ and then “added $f(t)$ to both sides” to get $2f(t)=0$. The confusion: it can look like we claimed $-f(t)+f(t)=f(t)$. We did not. The two sides get the same treatment but produce different results.

Watch each side separately

Adding the same thing to both sides of an equation keeps it true. Do it and simplify each side on its own:

• LLeft side: it was $f(t)$, we add $f(t)$, and two copies of the same thing is twice the thing: $2f(t)$. This is where the 2 comes from.
• RRight side: it was $-f(t)$, we add $f(t)$, and anything plus its own negative cancels to $0$. This is where the 0 comes from. (Never $f(t)$: that would be $-1+1=1$, which is false.)

Same move with a plain x

If the $f(t)$ clutter is the problem, rename it: let $x=f(t)$, one unknown number. The proof line is then just ordinary equation-solving:

Sanity check with a number: could $x=5$ satisfy $x=-x$? That would need 5 = −5, false. Could $x=0$? 0 = −0 is true. Zero is the only number equal to its own negative, which is exactly what the algebra concluded, and (applied at every t) why the only both-odd-and-even function is the zero function.

Yes, but only one function in all of mathematics pulls it off: the zero function, $f(t)=0$ for every $t$ (the flat line lying exactly on the horizontal axis). Nothing else qualifies. Here is why.

The proof: force both rules at once

Suppose some function $f$ is BOTH. Then it must obey both defining equations at the same time:

Both equations describe the SAME quantity $f(-t)$ on their left, so their right-hand sides must be equal to each other:

Now add $f(t)$ to both sides:

And this holds for every $t$. So the only way to be both is to be zero everywhere. The single number that is its own negative is $0$, and a function that must equal its own negative at every point has no choice but to be flat zero.

The picture: the one shape both symmetries allow

The flat line on the axis is the only graph that survives BOTH symmetry operations. Mirror it left-to-right, unchanged. Spin it 180° about the origin, still unchanged. Any bump anywhere would break one of the two.

The takeaway on the three labels

• 1“Odd” and “even” are NOT opposites. Most functions are neither (like our square wave), some are one, and exactly one, $f=0$, is both.
• 2So “both” is a real but trivial case. If a symmetry argument ever forces your function to be both odd and even, it is telling you the function is identically zero, sometimes a useful sanity check that a coefficient must vanish.

Our unit square wave is nowhere near this: it sits at 1 half the time, so it is not the zero function, and (as shown in the entries below) it is neither odd nor even.

“Odd” and “even” are two specific kinds of symmetry a graph can have. A function can be one, the other, or neither. To decide, you run two tests. If both fail, the verdict is “neither”, which is what happened here.

What the two symmetries look like

Before testing, picture what you are testing FOR. These are the only two special mirrors that matter:

In symbols, comparing the height at $t$ with the height at $-t$:

The two heights we already know

From reading the wave earlier, we have a value and its mirror-partner:

• +$h(\frac{\pi }{2})=1$  (that time is in the ON block)
• −$h(-\frac{\pi }{2})=0$  (its mirror lands in the OFF block)

To DISPROVE a symmetry you only need one place where it fails, so this single pair of heights is enough to settle both tests.

Test 1 — is it EVEN?

Even demands $h(-t)=h(t)$. Put our two heights in:

$0$ is not $1$, so the even equation is false. Not even.

Test 2 — is it ODD?

Odd demands $h(-t)=-h(t)$. That would need $h(-\pi /2)$ to equal $-1$:

$0$ is not $-1$, so the odd equation is false too. Not odd.

Verdict

Even test: failed. Odd test: failed. A function that is neither even nor odd is called, simply, neither. That is the conclusion.

Two ways to see it instantly, no test point needed

• 1Can’t be odd: an odd graph must dip below the axis (every positive bit needs a matching negative bit rotated through the origin). This wave only ever sits at $0$ or $1$, never negative, so odd is impossible on sight.
• 2Can’t be even: even means a left/right mirror about the vertical axis. The ON block sits to the RIGHT of $0$ with nothing mirroring it on the left (that side is OFF), so the picture isn’t mirror-symmetric.

Why we bothered checking

Symmetry is a shortcut: an even function needs only cosine terms, an odd function only sine terms, and either one kills half the coefficients before you integrate. “Neither” means no free lunch from symmetry directly, BUT this wave is secretly (odd wave) + (constant), so once you subtract its mean it becomes odd, and the series is a DC term plus sines only. That is why checking symmetry first, even when the answer is “neither”, still pays off.

Short answer: yes, that particular 2 is arbitrary. But this is worth pinning down properly, because a maths page is full of numbers and some are free choices while others are locked by the problem. Telling them apart stops a lot of confusion.

The 2 in π/2 was MY choice (a test point)

Remember where $\pi /2$ came from: to check whether the square wave was odd or even, I needed to pick some time and read the wave’s height there and at its mirror. I picked $\pi /2$. Nothing forced that. I just wanted a convenient spot sitting safely inside the ON block $(0,\pi )$, away from the jumps at the edges. Any time in that block would have done the job:

To prove it is free, redo the whole odd/even test with t = 1 instead:

• 11 is in the ON block (since $1<\pi \approx 3.14$), so $h(1)=1$.
• 2Its mirror: $h(-1)=h(-1+2\pi )=h(5.28)=0$ (OFF block).
• 3$1\ne 0$ and $1\ne -0$, so “neither odd nor even.” Same verdict as π/2 gave.

The conclusion did not depend on the number I picked, which is exactly what “arbitrary” means here.

But most 2’s in Fourier are NOT free, they are forced

Do not carry the “it’s just a number” lesson too far. The other 2’s floating around this topic are locked by the maths and cannot be changed:

• πThe 2 in $\omega =\frac{2\pi }{T}$: forced. It is there because one full turn of a circle is $2\pi$ radians. Change it and $\omega$ is simply wrong.
• ½The 2 in the DC term $\frac{a_0}{2}$ and the $\frac{2}{T}$ in every coefficient: forced. They fall out of the orthogonality integrals (a ${\sin }^2$ or ${\cos }^2$ averages to $T/2$ over a period). That $T/2$ is what the $\frac{2}{T}$ cancels.

The rule of thumb

• ?Ask: if I swapped this number, would the answer still be right?
• ✓Yes → it was a free CHOICE (a test point, a convenient limit of integration, a sample value). Pick whatever is easiest.
• ✗No → it is FORCED by a definition or an identity (a full circle is $2\pi$, orthogonality gives $T/2$). Do not touch it.

So: the 2 in the test point π/2 is a free choice, and swapping it for 1 or π/4 changes nothing. The 2 in ω = 2π/T and in a₀/2 is forced, and swapping it breaks the maths. Both are “a 2”, but they are doing completely different jobs.

No trig, no integration here. This is just: you are handed a time, and you have to read off the wave’s height at that time. We go one tiny step at a time.

First: what does “h(t)” even mean?

Think of $h$ as a machine. You feed it a time, it gives you back a height. That is all the notation $h(t)$ means: “the height of the wave at time $t$.”

So what is $h$ on its own? It is simply the name of the wave, the way a person is named Tom. $h$ by itself is not a number, it is the whole rule/shape. The letter is arbitrary (could be $f$, $g$, $v$); here it is $h$, handy for height. When you bolt a time onto it, $h(1.57)$, THAT is a number: the height the wave named $h$ reaches at that time.

Wait — π/2 is horizontal, 1 is vertical. How can they be equal?

They are not equal, and this is the single most important thing to get straight. $h(\pi /2)=1$ is NOT saying “π/2 is the same as 1.” They live on two different axes and never get compared:

• ↔π/2 is a horizontal thing: a time, a spot along the floor. It answers “how far along?”
• ↕1 is a vertical thing: a height, how tall the wave stands. It answers “how high?”

The machine $h$ is the bridge between the two: you hand it a floor-position (the time), and it hands back a height. So $h(\pi /2)=1$ reads: “at the floor-position π/2, the wave is standing 1 tall.” The left side is a height, that is why the answer is a height.

Analogy: a car’s speedometer. “At 3 o’clock the speed was 50 mph” does not mean 3 = 50. The 3 is a time, the 50 is a speed; the speedometer links them. Same here: the time π/2 and the height 1 are different kinds of thing, and $h$ is what links them.

Second: this machine’s rule is a light switch on a timer

Our wave’s rule has just two settings. Written out, it says:

• 1For any time between 0 and π  →  the height is 1 (switch ON).
• 2For any time between π and 2π  →  the height is 0 (switch OFF).

Here is that rule drawn on a timeline. The blue stretch is where the height is 1, the grey stretch is where it is 0:

Third: turn π/2 into a normal number

$\pi$ is just a number, about 3.14. So $\pi /2$ means 3.14 shared into 2, which is about 1.57. Nothing more mysterious than that:

So the question “what is $h(\pi /2)$?” is really just “what is the height at time 1.57?”

Fourth: find where 1.57 sits on the timeline

Put 1.57 on the same timeline. Is it in the blue (ON) stretch or the grey (OFF) stretch? The ON stretch runs 0 to 3.14, and 1.57 is smaller than 3.14, so it lands inside the blue stretch, roughly in the middle of it:

Fifth: read off the height for that stretch

The blue stretch has height 1 everywhere. Time 1.57 is in the blue stretch. So the height at 1.57 is 1. Drawn on the actual wave, you go up from 1.57 on the bottom axis until you hit the wave, and you hit it at height 1:

Do it once more, so it sticks: h(3π/2)

Same four steps. Turn it into a number: $3\pi /2$ is $3\times 3.14/2\approx 4.71$. Where does 4.71 sit? It is bigger than 3.14, so it is past π, in the grey OFF stretch (3.14 to 6.28). The grey stretch has height 0. So:

That is the exact value used for h(−π/2) in the odd/even test: −π/2 is the same spot as 3π/2 once you slide over by one full period, and both read 0.

The whole thing in one sentence

• →Turn the time into a plain number.
• →See which stretch (ON or OFF) that number falls in.
• →Read off that stretch’s height. Done.

There is genuinely no calculation, it is a lookup. The wave only ever sits at 1 or 0, and which one you get depends only on whether your time landed before π or after it.

The setup: one period of the signal is $h(t)=1$ on $0<t<\pi$ and $h(t)=0$ on $\pi <t<2\pi$, repeated forever. A switch that is ON half the time, OFF half the time. Period $T=2\pi$, so $\omega =\frac{2\pi }{T}=1$ rad/s.

Part 1: odd, even, or neither, tested properly

The definitions are about comparing the function at $t$ and at $-t$:

Geometrically: even means the graph is a mirror image in the vertical axis; odd means the graph is unchanged by a 180° rotation about the origin. To DISPROVE a symmetry you only need one point where it fails, so the practical test is: pick a convenient $t$, read the height there and at $-t$, compare.

Take $t=\frac{\pi }{2}$. From the graph, $h(\frac{\pi }{2})=1$ (inside the ON block). For the height at $-\frac{\pi }{2}$, use periodicity, shifting by one whole period changes nothing:

because $\frac{3\pi }{2}$ lies in the OFF block $(\pi ,2\pi )$. Now run both tests at this one point:

• 1Even test: is $h(-t)=h(t)$?  Here $0\ne 1$. Fails. Not even.
• 2Odd test: is $h(-t)=-h(t)$?  Here $0\ne -1$. Fails. Not odd.

Both symmetries fail, so the wave is neither. There is also a faster argument against oddness that needs no point at all: an odd function must take the value $-h(t)$ whenever it takes $h(t)$, i.e. it needs matching NEGATIVE values. This wave only ever equals $0$ or $1$, it never goes below the axis, so it cannot possibly be odd.

The hidden symmetry: "neither" here means "odd plus an offset"

Here is the insight that explains the final answer before you compute anything. The wave itself has no symmetry, but subtract its average height $\frac{1}{2}$ and look at what is left:

$h(t)-\frac{1}{2}$ is a $\pm \frac{1}{2}$ square wave, and THAT function is genuinely odd. So the signal is really (odd function) + (constant). Odd functions expand in sines only, constants are the DC term, and that is the entire structure of the answer: a constant $\frac{1}{2}$ plus sines, with every $a_n$ destined to be zero. The calculation below just confirms it.

Part 2: what the DC term is

"DC" is borrowed from electronics: direct current, the steady, zero-frequency part of a signal, what a battery supplies, as opposed to the wiggling AC part. In a Fourier series the DC term is the constant out front, and it always equals the mean value of the signal over one period:

You can see the answer with no integration: the wave is at height $1$ for exactly half of every period and at $0$ for the other half, so its average height is $\frac{1}{2}$. That is the dashed line in the first figure, the level the wave "balances" around. Now the formula version, which is what the tutorial writes:

• 1The integral shrank from $(0,2\pi )$ to $(0,\pi )$ because the function is ZERO on the second half, integrating 0 adds nothing.
• 2Watch the factor of two: the course convention writes the constant as $\frac{a_0}{2}$, so $a_0=1$ but the DC level is $\frac{1}{2}$. Writing $1$ as the constant term is THE classic lost mark.
• 3Engineering check: this is a 50% duty-cycle switch on a 1 V rail, and its DC level is duty × amplitude = ½. The same idea runs PWM in Power Electronics: vary the duty cycle and you vary the DC you deliver.

Part 3: the remaining coefficients, briefly

The cosine coefficients die because $\sin (n\pi )=0$ at both limits (as the hidden-odd argument predicted):

The sine coefficients survive, split by the $\cos (n\pi )={(-1)}^n$ even/odd flip (see the entry below on exactly this):

Assemble constant + surviving sines:

Read the answer back against the structure we predicted: the $\frac{1}{2}$ is the DC term (the mean), and everything else is sines because the wave minus its mean is odd. Only odd harmonics appear (1st, 3rd, 5th…), the coefficients decay like $\frac{1}{n}$ because the wave has jumps, and at each jump the series converges to the midpoint $\frac{1}{2}$, which is, once again, the DC level.

The whole split is powered by one fact: $\cos (n\pi )$ is always either +1 or −1, flipping as $n$ steps through the integers. Plug those two values into $-\frac{\pi \cos (n\pi )}{n}$ and you get the two cases.

Step 1: cos(nπ) lands only on ±1

Walk the cosine graph at whole multiples of $\pi$. Every step of $\pi$ is half a wavelength, so you jump from one peak straight to the opposite one:

Even $n$ gives $+1$, odd $n$ gives $-1$. That is exactly what the compact identity says:

Step 2: substitute each value

Everything except the cosine, the $-\frac{\pi }{n}$, just rides along. Only the $\cos (n\pi )$ changes sign:

• E$n$ even, $\cos (n\pi )=+1$:  $-\frac{\pi }{n}·(+1)=-\frac{\pi }{n}$
• O$n$ odd, $\cos (n\pi )=-1$:  $-\frac{\pi }{n}·(-1)=+\frac{\pi }{n}$

The odd case is the one to watch: the two minus signs cancel (the leading $-$ times the $-1$ from the cosine), flipping the result positive. That sign flip is the entire content of the brace:

The one-line version

You do not have to write a brace at all. Because $\cos (n\pi )={(-1)}^n$, the same result folds into a single alternating term:

The bumped exponent, $n+1$ instead of $n$, is precisely that "minus signs cancel" flip written algebraically: $-{(-1)}^n={(-1)}^{n+1}$. Both forms are the same number; use the brace when a sketch or a physical sign matters, the ${(-1)}^{n+1}$ form when you are about to sum the series.

Where this shows up: it is the boundary term from $\int t\sin (nt)dt$ in the sawtooth $b_n$ coefficient. That is why the sawtooth series alternates in sign, $+,-,+,-$: the ${(-1)}^{n+1}$ is doing it.

Integration by parts is the product rule run backwards. It trades the integral you cannot do for one you can, and it is the tool behind every $\int t\sin (nt)dt$ and $\int t\cos (nt)dt$ in the Fourier unit.

The formula

With limits, the $uv$ term gets evaluated at them too:

Where it comes from: the product rule, reversed

Differentiate a product of two functions:

Integrate both sides with respect to $t$. The left side is a derivative being integrated, so it collapses back to $uv$:

Move one integral to the other side and you have the formula. Nothing deeper than that: by parts IS the product rule, read right-to-left.

How to pick u and dv

• 1$u$ = the part that gets simpler when differentiated. Powers of $t$ are the classic pick: $t$ differentiates to $1$ and vanishes from the leftover integral.
• 2$dv$ = the part you can integrate on sight: $\sin (nt)$, $\cos (nt)$, $e^{kt}$ all recycle forever under integration.
• 3Fourier habit: always $u=t$ (or $t^2$), $dv=$ the trig part. A $t^2$ needs by parts twice; each pass drops the power by one and pulls out a factor of $\frac{1}{n}$.

The workhorse, in full: ∫ t sin(nt) dt

Choose $u=t$ and $dv=\sin (nt)dt$. Then $du=dt$, and integrating $dv$ gives $v=-\frac{\cos (nt)}{n}$. Feed everything into the formula:

(The two minus signs on the leftover integral cancelled into a plus.) Finish with $\int \cos (nt)dt=\frac{\sin (nt)}{n}$:

The companion result, same moves with $dv=\cos (nt)dt$:

Where marks die: the minus sign in $v=-\frac{\cos (nt)}{n}$ and the double negative it creates one line later. Also: no +c is needed for $v$ itself, any one antiderivative works, the constant is added once at the very end (and not at all in definite integrals).

They come from the definition of polar coordinates. Polar describes a point not by "how far right, how far up" $(x,y)$, but by "how far from the origin, at what angle" $(r,\theta )$. The cos and sin are just the conversion between the two languages.

Step 1: the conversion formulas (SOH-CAH from the triangle)

Drop a vertical from the point down to the x-axis: you get a right triangle with hypotenuse $r$ and angle $\theta$ at the origin. The horizontal leg is $x$ (adjacent), the vertical leg is $y$ (opposite). Ordinary trigonometry:

Multiply each through by $r$ and you get the two substitution formulas:

Step 2: substitute both into x² + y²

(Squaring a product squares each factor: ${(r\cos \theta )}^2=r^2{\cos }^2\theta$.)

Step 3: factor out r² and use the Pythagorean identity

Both terms share $r^2$, so pull it out, and the bracket that remains is the most famous identity in trigonometry, equal to 1 for every angle:

Why ${\cos }^2\theta +{\sin }^2\theta =1$: it is Pythagoras on the same triangle. The legs are $r\cos \theta$ and $r\sin \theta$, the hypotenuse is $r$, so legs² sum to hypotenuse², and dividing through by $r^2$ leaves exactly that identity.

So the whole line in the notes is: substitute the polar definitions of $x$ and $y$, square them, factor, and the trig disappears, leaving the beautifully simple integrand $r^2$. That collapse is precisely why polar is the right tool whenever $x^2+y^2$ appears.

Because $25$ is $5^2$, and for a circle centred at the origin the number on the right of $x^2+y^2$ is the radius squared. So you read the radius off by square-rooting it.

Where $x^2+y^2=R^2$ comes from

A circle is every point at the same distance $R$ from the centre. The distance of a point $(x,y)$ from the origin is, by Pythagoras, $\sqrt{x^2+y^2}$. Set that equal to $R$ and square both sides:

So match it to 25

The region's boundary is $x^2+y^2=25$. Line that up with $x^2+y^2=R^2$:

(You take the positive root because a radius is a length.)

Why that instantly gives the polar limit

In polar, $r^2=x^2+y^2$, so the filled disk $x^2+y^2\le 25$ is just:

General rule: on a circle centred at the origin, $x^2+y^2=c$ is a circle of radius $\sqrt{c}$. The bare number is always the radius SQUARED, so square-root it. (Here $\sqrt{25}=5$; if it had been $9$ the radius would be $3$.)

Because once the inner integral is finished, $\frac{1}{3}{R}^3$ is a plain constant as far as $\theta$ is concerned: $R$ is a fixed radius, and there is no $\theta$ left in it. And a constant slides straight out of the integral.

The mechanics

Pull the constant to the front:

and the leftover integral is just integrate 1 across the interval, which measures the interval's width:

So the whole thing is the constant times $2\pi$:

The one-line picture

An integral is the area under a graph. Plot the constant height $\frac{1}{3}{R}^3$ against $\theta$ from $0$ to $2\pi$ and you get a flat line, so the region under it is simply a rectangle. No calculus needed once you see that.

Contrast: this shortcut works ONLY because $\theta$ dropped out of the integrand. If a $\theta$ had remained, say $\int \theta d\theta =\frac{{\theta }^2}{2}$, integrating would reshape it and you could not just multiply. "Contains no $\theta$" is the flag that the collapse to a single factor of $2\pi$ is allowed.

First, what substitution even is: it is the chain rule run backwards. When an integral contains some inner expression AND (a multiple of) that inner expression's derivative, you rename the inner expression $u$, let $du$ swallow the derivative part, and you are left with a clean, easy integral in $u$.

Step 0: the piece we are integrating

Only the inner (the $r$) integral needs work; $\theta$ comes later. It is:

Step 1: why u = R² − r² is the right pick

Look at what sits inside the square root, $R^2-r^2$, and differentiate it (remember $R$ is a fixed constant, so $R^2$ differentiates to 0):

That derivative is $-2r$, and there is an $r$ already multiplying the root out front. So the $rdr$ in the integral is exactly the derivative part, up to the constant $-2$. That match is the whole reason this substitution works.

Step 2: build the substitution

Set $u$ to the inner expression, differentiate, then rearrange to isolate the exact clump $rdr$ that appears in the integral:

That last arrow is just dividing $du=-2rdr$ by $-2$ on both sides.

Step 3: rewrite the whole integral in u

Two swaps: the root $\sqrt{R^2-r^2}$ becomes $\sqrt{u}$, and the clump $rdr$ becomes $-\frac{1}{2}du$. The constant $-\frac{1}{2}$ then just slides out front:

($\sqrt{u}$ is the same as $u^{1/2}$, written that way so the power rule applies.)

Step 4: integrate u^1/2 with the power rule

The power rule $\int u^{n}du=\frac{u^{n+1}}{n+1}$ with $n=\frac{1}{2}$ gives $n+1=\frac{3}{2}$:

Dividing by $\frac{3}{2}$ is the same as multiplying by $\frac{2}{3}$, which is where the $\frac{2}{3}$ comes from.

Step 5: combine the constants and undo the substitution

$\frac{1}{2}·\frac{2}{3}=\frac{2}{6}=\frac{1}{3}$, and we put $u$ back as $R^2-r^2$ so we can use the original $r$ limits. That is the antiderivative the notes jumped straight to.

Step 6: put in the limits r = 0 and r = R

Evaluate $-\frac{1}{3}{(R^2-r^2)}^{3/2}$ at the top then the bottom, and subtract.

• RAt $r=R$: $R^2-R^2=0$, and $0^{3/2}=0$, so this end is 0.
• 0At $r=0$: $R^2-0=R^2$, and ${(R^2)}^{3/2}=R^{2·3/2}=R^3$, so this end is $-\frac{1}{3}{R}^3$.

Definite integral is top minus bottom. Watch the double negative:

Subtracting a negative flips it to a plus, which is why the answer comes out positive.

Step 7: the outer integral, then double it

$\frac{1}{3}{R}^3$ has no $\theta$ in it, so integrating over $\theta$ from $0$ to $2\pi$ just multiplies it by $2\pi$:

That is one half (a hemisphere). Double it for the full sphere:

The one habit that makes substitution click: scan the integrand for an inner function whose derivative (up to a constant) is also sitting there. Name it $u$, let $du$ absorb that derivative, and what remains is always a plain power of $u$ you can integrate on sight.

The strip is vertical, so $y$ is the inner variable. Fix an $x$ and slide up the strip: the lower limit is whichever boundary the strip sits on (the curve underneath), and the upper limit is whichever boundary it hits at the top (the curve above). So the whole question is: between $x=1$ and $x=2$, which line is higher, $y=3x-2$ or $y=x$?

Fastest way: test one x

Both lines meet at the corner $(1,1)$, so pick any $x$ strictly inside the strip, say $x=\frac{3}{2}$, and read off both heights:

and $\frac{5}{2}>\frac{3}{2}$, so $3x-2$ is the higher one. That makes it the top (upper limit) and $y=x$ the bottom (lower limit).

Why it stays that way across the whole strip

You do not have to re-test every point, the slopes settle it:

• 1Both boundary lines pass through the same corner $(1,1)$.
• 2$y=3x-2$ has slope 3; $y=x$ has slope 1. From a shared point the steeper line climbs faster, so for every $x>1$ it sits above the gentler one.

So $3x-2$ is on top for the entire strip $1\le x\le 2$, never crossing below until the two lines have already met at the apex.

Hence the inner integral runs bottom to top:

The rule to carry away: for a vertical strip the upper limit is whichever curve is on top and the lower limit is whichever is underneath. When two curves swap or you are unsure, plug one $x$-value into both, the larger $y$ is the ceiling.

This uses vertical strips: $y$ is the inner variable (each strip runs bottom to top) and $x$ is the outer variable (the strips sweep left to right). Every strip starts on the same bottom edge, but the top edge changes partway across, and that is the whole reason for the two integrals added together.

Why the + (why it splits at x = 2)

Walking left to right, the bottom of every strip is the same line, $y=x$ (the green edge). The top is not:

• LFor $1\le x\le 2$ the top edge is the blue line $y=3x-2$.
• RFor $2\le x\le 3$ the top edge is the orange line $y=-x+6$.

A single integral needs one formula for its upper limit, but the top switches at the apex $(2,4)$. So you cut the region at $x=2$, integrate each half with its own top line, and add. That is the "done twice" you spotted.

Where each limit comes from

• 1Inner lower $y=x$ — the bottom of every strip, the green edge joining $(1,1)$ and $(3,3)$. Same in both integrals.
• 2Inner upper $3x-2$ (left integral) — the blue top edge, valid only while $x\le 2$.
• 3Inner upper $-x+6$ (right integral) — the orange top edge, valid once $x\ge 2$.
• 4Outer $x$: $1\to 2$ then $2\to 3$ — the x-span of each half: left vertex $x=1$, apex $x=2$ (the cut), right vertex $x=3$.

Seam check: at $x=2$ the blue top gives $3\left(2\right)-2=4$ and the orange top gives $-2+6=4$, both the apex, so the two halves meet cleanly. At $x=1$ and $x=3$ the top equals the bottom, so the strips shrink to the side vertices.

Each edge of the triangle is the straight line through two of its corners, and two points fix a line completely. The recipe is always the same two steps: find the slope, then feed it and one point into point-slope form.

Blue edge: (1, 1) to (2, 4)

Green edge: (1, 1) to (3, 3)

Orange edge: (2, 4) to (3, 3)

Pairing check: the three corners give exactly three pairs, and each pair is one edge, (1,1)–(2,4), (1,1)–(3,3) and (2,4)–(3,3). The dashed line $x=2$ in the moodle figure is not an edge; it is the vertical through the apex (2,4), marking where the top boundary switches from the blue line to the orange line, which is why that integral has to be split there.

Two integral signs means four numbers to justify. Every one is read straight off the triangle $R$, whose edges are $x\ge 0$, $y\ge 0$ and $x+y\le 1$. We integrate in horizontal strips: $x$ is the inner variable (moving along a strip), $y$ is the outer variable (choosing which strip).

Inner integral — the two x-limits

Fix a height $y$ and slide across the strip. The limits are where the strip enters and leaves the region:

• 1$x=0$  (lower) — the strip enters on the left edge, the y-axis. That edge is the boundary $x\ge 0$.
• 2$x=1-y$  (upper) — the strip leaves on the slanted edge $x+y=1$. Solve that for $x$: $x=1-y$. It carries a $y$ only because the edge is sloped.

Outer integral — the two y-limits

Now sweep the strip through every height the triangle occupies. These bounds must be constants:

• 3$y=0$  (lower) — the lowest strip lies on the bottom edge, the x-axis, which is the boundary $y\ge 0$.
• 4$y=1$  (upper) — the highest strip shrinks to the apex $(0,1)$, where the left edge $x=0$ meets the slant $x+y=1$.

The one rule that fixes the order: inner limits describe the moving strip and may contain the outer variable (that is the $1-y$), while outer limits describe the whole region and are always plain constants ($0$ and $1$).

Every limit comes from reading the three edges of the triangle and then following one horizontal strip. The region $R$ is $x\ge 0$, $y\ge 0$, $x+y\le 1$, so its edges are:

• 1$x=0$  — the y-axis (left edge)
• 2$y=0$  — the x-axis (bottom edge)
• 3$x+y=1$, rearranged to $x=1-y$  — the slanted hypotenuse (right edge)

Inner limits: follow one horizontal strip

Horizontal strips means $x$ is the inner variable. Fix a height $y$ and move along the strip left to right, asking where it enters and exits the region:

It enters on the left at $x=0$ (the y-axis), and exits on the right at the hypotenuse. Solving $x+y=1$ for $x$ gives $x=1-y$. So the inner limits are

The upper one depends on $y$ precisely because the right edge is slanted, not vertical: as the strip rises, its right end slides left along the line.

Outer limits: sweep the strip over the region

The outer variable $y$ runs over every height the region reaches, and these must be constants. The lowest strip sits on the base at $y=0$; the highest is where the triangle pinches to its apex, where $x=0$ meets $x+y=1$, i.e. $y=1$. So $y:0\to 1$, and the whole integral is

The two rules to carry away

• 1Inner limits describe the strip and may depend on the outer variable (here the right limit is $1-y$). Read them off where the strip enters and exits.
• 2Outer limits describe the whole region and must be constants (here $y$ from $0$ to $1$). Read them off the lowest and highest the outer variable reaches.

Sanity check on the slant: at $y=0$ the strip runs $x:0\to 1$ (the full base); at $y=1$ it runs $x:0\to 0$ (zero length, the apex). That shrinking is exactly what $x=1-y$ encodes. Flip to vertical strips and the same logic gives $y:0\to 1-x$ inner, $x:0\to 1$ outer.

It’s a quick check that the direction vector really has length 1, and the two fractions are just its components squared, nothing more.

The line just above it sets the unit direction from the angle $a=\frac{\pi }{3}$:

The length of any 2D vector $(a,b)$ is $\sqrt{a^2+b^2}$, so here:

and that is exactly where the two numbers come from:

${(\frac{1}{2})}^2=\frac{1}{4}$  — the first component squared, since $\cos \frac{\pi }{3}=\frac{1}{2}$.

${(\frac{\sqrt{3}}{2})}^2=\frac{3}{4}$  — the second component squared: $\sqrt{3}$ squared is 3, over $2^2=4$.

Why the check is there at all

The directional-derivative formula $\nabla f·\hat{\mathbf{u}}$ only gives the true slope if $\hat{\mathbf{u}}$ has length exactly 1. This line verifies that before it’s used. And it will always come out to 1 for a direction written $(\cos a,\sin a)$, because $\frac{1}{4}+\frac{3}{4}$ is really ${\cos }^{2}a+{\sin }^{2}a$, and the Pythagorean identity guarantees:

So it is partly a genuine check, and partly a reassurance that the $(\cos a,\sin a)$ trick automatically produces a unit vector.

Picture the function’s graph as a hilly landscape: at every point $(x,y)$ the value $f(x,y)$ is the height of the ground. Standing at one spot, you can step off in any compass direction, and depending which way you face the ground rises faster, rises slower, stays level, or drops.

The direction of steepest ascent is the compass direction you’d face to climb uphill as fast as possible. Turn slightly away and the ground still rises, but less steeply; face exactly opposite and you get steepest descent.

It is exactly the gradient

That best direction is given directly by the gradient, the vector of the partial derivatives you were computing:

The direction this vector points (in the flat $xy$-plane) is the direction of steepest ascent. Its length tells you how steep that steepest climb is, the slope in the best direction:

Why the partials assemble into the right direction

$\frac{\partial f}{\partial x}$ is how fast height changes if you walk purely east, $\frac{\partial f}{\partial y}$ if you walk purely north. Any real direction is a blend of the two, and the slope you feel along a unit direction at angle $\alpha$ is the directional derivative, the dot product of the gradient with that direction:

A dot product is largest when the two vectors are parallel, so the slope is biggest exactly when your walking direction lines up with $\nabla f$. That is the whole reason the gradient is the steepest direction.

Two facts to carry into the exam

• 1The gradient is always perpendicular to the level curve (the contour of constant height) through your point. Along a contour the height doesn’t change, and steepest-uphill is exactly at right angles to “no change”.
• 2Steepest descent is just $-\nabla f$, the same arrow flipped. That’s the idea behind gradient-descent: to reach a minimum, keep stepping in the $-\nabla f$ direction.

A quick concrete case

Take a bowl $f(x,y)=x^2+y^2$. Its gradient is

At any point this arrow points straight away from the origin, which is indeed the steepest way up the side of the bowl, and $-\nabla f$ points back down toward the bottom.