Where does the factor \frac12r^2\,\delta\theta come from?

Area of a thin sector of radius r and angle \delta\theta.

What is the polar convention used for r?

r\ge0, so each \theta gives one point.

Identity for \cos^2\theta?

\cos^2\theta=\frac12(1+\cos2\theta)

Identity for \sin^2\theta?

\sin^2\theta=\frac12(1-\cos2\theta)

Limits for the whole of r=4\cos\theta with r\ge0?

-\frac{\pi}{2} to \frac{\pi}{2}

First step with \frac12\int(a+b\cos\theta)^2\,d\theta?

Expand the bracket, then replace \cos^2\theta using the double-angle identity.

How do you find where two polar curves meet?

Equate the two expressions for r and solve for \theta (also check the pole).

How do you find the area inside two curves?

Split at the intersection and use the curve with the smaller r on each range.

When can you double an area from 0 to \pi?

When the curve is symmetrical about the initial line.

Area of the region bounded by r=2\theta from \theta=0 to \pi?

\frac12\int0^{\pi}4\theta^2\,d\theta=\frac{2\pi^3}{3}

Area in polar coordinatesEdexcel International A Level Further Maths: Flashcards

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Area bounded by $r=f(\theta)$ between $\theta=\alpha$ and $\theta=\beta$?

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Area bounded by $r=f(\theta)$ between $\theta=\alpha$ and $\theta=\beta$ ?: $A=\frac12\int_\alpha^\beta r^2\,d\theta$
Where does the factor $\frac12r^2\,\delta\theta$ come from?: Area of a thin sector of radius $r$ and angle $\delta\theta$ .
What is the polar convention used for $r$ ?: $r\ge0$ , so each $\theta$ gives one point.
Identity for $\cos^2\theta$ ?: $\cos^2\theta=\frac12(1+\cos2\theta)$
Identity for $\sin^2\theta$ ?: $\sin^2\theta=\frac12(1-\cos2\theta)$
$\int\cos2\theta\,d\theta$ ?: $\frac12\sin2\theta+c$
Limits for the whole of $r=4\cos\theta$ with $r\ge0$ ?: $-\frac{\pi}{2}$ to $\frac{\pi}{2}$
Area enclosed by $r=a(1+\cos\theta)$ ?: $\frac{3\pi a^2}{2}$
First step with $\frac12\int(a+b\cos\theta)^2\,d\theta$ ?: Expand the bracket, then replace $\cos^2\theta$ using the double-angle identity.
How do you find where two polar curves meet?: Equate the two expressions for $r$ and solve for $\theta$ (also check the pole).
How do you find the area inside two curves?: Split at the intersection and use the curve with the smaller $r$ on each range.
When can you double an area from $0$ to $\pi$ ?: When the curve is symmetrical about the initial line.
Area of the region bounded by $r=2\theta$ from $\theta=0$ to $\pi$ ?: $\frac12\int_0^{\pi}4\theta^2\,d\theta=\frac{2\pi^3}{3}$

Exam questions on Area in polar coordinates

The curve $C$ has polar equation $r=4\cos\theta$ , for $-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}$ .
Find the exact area of the region bounded by $C$ and the half-lines $\theta=0$ and $\theta=\frac{\pi}{4}$ .2 marks
The spiral $S$ has polar equation $r=2\theta$ , for $\theta\ge0$ .
Find the exact area of the region bounded by $S$ and the half-lines $\theta=\pi$ and $\theta=2\pi$ .2 marks
The curve $C$ has polar equation $r=3+2\cos\theta$ , for $0\le\theta\le2\pi$ .
Show that the area enclosed by $C$ is $11\pi$ .3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).