MATH U101 · Multivariable Calculus · Module 1

Polar Coordinates

Thomas' Calculus (15th ed.) §11.3 §11.4* §11.5 — a new way to name points, the curves it makes easy, and how to measure area and length in it.

Start here — a fact worth knowing

Polar coordinates are not in the CBSE 11th/12th syllabus. Almost everyone in the lecture hall — BITSAT or SAT channel — is meeting this topic for the first time. The only background this module leans on is trigonometry: unit-circle values, the graphs of sin and cos, and (for §11.5) one identity and basic integration. That's the whole entry fee.

1 · The idea: name a point by distance and direction

Cartesian coordinates answer "how far right, how far up?" Polar coordinates answer a different, often more natural pair of questions: how far from the origin, and at what angle?

Fix the origin O (now called the pole) and the positive x-axis (now the initial ray). A point gets coordinates (r, θ):

r = directed distance from the pole; θ = angle from the initial ray (radians, counterclockwise positive).

Two features make polar coordinates behave differently from Cartesian — and both are exam favourites:

Names are not unique

The same point has infinitely many polar addresses. (2, π/6), (2, π/6 + 2π), and (−2, π/6 + π) are all the same point. In Cartesian coordinates a point has exactly one name; in polar it has infinitely many. Every "find the intersections of two curves" problem turns on this.

Negative r means "walk backwards"

(−2, π/6) means: face the direction π/6, then walk 2 units the opposite way — you land in the third quadrant. Negative r is legal and routinely appears when graphing.

2 · Converting between polar and Cartesian §11.3

Four equations do all the work. The first two convert polar → Cartesian; the second two go back.

x = r cos θ y = r sin θ r² = x² + y² tan θ = y/x

Worked example 1Identify the curve r = 2 cos θ

The equation mixes r and θ. To use the conversion table we want r² and r cos θ, so multiply both sides by r: r² = 2r cos θ.
Substitute: x² + y² = 2x.
Complete the square: (x − 1)² + y² = 1 — a circle of radius 1 centred at (1, 0).

The "multiply by r" move is the single most-used trick in this section. Memorise the outcome too: r = 2a cos θ is always a circle of radius a through the pole, sitting on the x-axis (with sin, on the y-axis).

Classic trap

tan θ = y/x does not pin down θ by itself — tan has period π, so it can't tell quadrant II from IV. When converting a point like (x, y) = (−1, 1), always sketch first: the point is in quadrant II, so θ = 3π/4, not −π/4.

3 · Graphing r = a ± b sin θ and r = a ± b cos θ §11.4*

This family is called the limaçons (lee-ma-sonn — French for snail). Your syllabus names this exact form, so expect it on quizzes. The whole game is: watch r grow and shrink as θ sweeps around.

The method that always works: a table of values

Take r = 1 + cos θ. March θ through the standard angles and record r:

θ	0	π/3	π/2	2π/3	π	4π/3	3π/2	5π/3	2π
r	2	3/2	1	1/2	0	1/2	1	3/2	2

Reading the story: the point starts 2 units out along θ = 0, spirals inward, touches the pole at θ = π, then swells back out. The heart-shaped result is a cardioid.

The shape depends only on the ratio a/b

All four shapes below are drawn exactly (these are computed curves, not sketches):

r = 1 + 2 sin θ
Inner loop (a/b < 1)

r = 1 + sin θ
Cardioid (a/b = 1)

r = 3 + 2 sin θ
Dimpled (1 < a/b < 2)

r = 5 + 2 sin θ
Convex (a/b ≥ 2)

Orientation is read straight off the equation: sin versions are symmetric about the y-axis, cos versions about the x-axis; a + bulges toward the positive axis, a − toward the negative.

Three shortcuts before you plot anything

Symmetry tests. If replacing θ → −θ leaves the equation unchanged (true for all cos versions), the curve is symmetric about the x-axis — plot the top half, mirror the rest. If θ → π − θ leaves it unchanged (all sin versions), symmetric about the y-axis.
Where is r largest/smallest? Just ask where sin or cos hits ±1. Maximum r = a + b, and the curve reaches the pole only if r = 0 has a solution (needs a ≤ b).
Solve r = 0 first. For r = 1 + 2 sin θ: sin θ = −½ at θ = 7π/6, 11π/6. Between those angles r is negative — that's exactly the inner loop, drawn "backwards" through the pole.

Classic trap

When r goes negative, students either skip those angles (losing the inner loop) or plot the point on the wrong side. Negative r points go in the opposite direction to θ. Also: keep your calculator in radians — a degrees-mode table produces garbage that looks plausibly like a spiral.

4 · Area and arc length in polar §11.5

Why this matters

In Cartesian, area under a curve is a sum of thin rectangles. In polar, the natural sliver is a thin pie slice of radius r and angle dθ, whose area is ½r² dθ (fraction dθ/2π of a full disc πr²). Summing slices gives the formula — understanding this derivation is worth more than memorising it.

A = 12 ∫_α^β r² dθ L = ∫_α^β √(r² + (dr/dθ)²) dθ

Worked example 2Area enclosed by the cardioid r = 1 + cos θ

The curve is traced once as θ runs 0 → 2π, so A = ½ ∫₀^2π (1 + cos θ)² dθ.
Expand: (1 + cos θ)² = 1 + 2 cos θ + cos² θ.
The only new tool needed: the half-angle identity cos² θ = (1 + cos 2θ)/2. You cannot integrate cos² without it — it appears in nearly every problem in this section.
Integrate over a full period: ∫ 1 = 2π, ∫ cos θ = 0, ∫ cos² θ = π.
A = ½ (2π + 0 + π) = 3π/2.

The trap that costs the most marks

Wrong θ-limits silently double (or halve) your area. The circle r = 2 cos θ is traced completely as θ runs from −π/2 to π/2. Integrate 0 → 2π and you trace it twice and get double the area. Before integrating, always ask: over which θ-interval is this curve drawn exactly once? Sketch first, integrate second — never the reverse.

Intersections: the pole is sneaky

To find where two polar curves meet, solving r₁(θ) = r₂(θ) is not enough, because the same point has many names. In particular both curves may pass through the pole at different values of θ — an intersection the algebra never shows. The reliable routine: solve the equation, and check the pole separately, and confirm against a sketch.

5 · Your minimal prerequisite kit

Everything this module assumes, in one list. If any line feels shaky, patch just that line — nothing else is needed:

Unit-circle values of sin and cos at 0, π/6, π/4, π/3, π/2 and their reflections in all quadrants — instant recall, in radians.
The graphs of sin θ and cos θ (to predict where r grows and shrinks).
Half-angle identities: cos²θ = (1+cos 2θ)/2, sin²θ = (1−cos 2θ)/2.
Completing the square (for identifying circles).
Basic integrals: ∫ cos kθ dθ, ∫ sin kθ dθ, and integrating constants.

6 · What to practise in Thomas

Odd-numbered exercises have answers in the back — do them in this order, checking as you go:

Skill	Where	How many
Plotting points, negative r, converting points & equations	§11.3 exercises	8–10
Graphing limaçons/cardioids from a table + symmetry	§11.4 exercises	6–8 (sketch every one by hand)
Areas inside one curve; area between two curves; arc length	§11.5 exercises	8–10, always sketching first

The starred §11.4* in the handout usually means the instructor covers it lightly or partially — but since the syllabus line names "graphing a ± b sin θ" explicitly, treat graphing as fully examinable and confirm with the instructor what the star excludes.