Area in polar coordinates

Warm-up

Three quick questions from earlier lessons.

Discover

A thin wedge of a disk of radius $r$ with angle $d\theta$ is almost a sector. A full disk ( $2\pi$ ) has area $\pi r^2$ . What is the area of the wedge?

$\tfrac{1}{2}r^2\,d\theta$
$r\,d\theta$

Key idea

Add up thin sectors.

The area swept by

r = f(\theta)

for

\alpha \le \theta \le \beta

is

A = \dfrac{1}{2}\displaystyle\int_{\alpha}^{\beta} r^2\,d\theta

.

Worked example

Area inside $r = 2\sin\theta$ , $0 \le \theta \le \pi$ .

A = \tfrac{1}{2}\displaystyle\int_0^{\pi} 4\sin^2\theta\,d\theta

r² = 4 sin²θ.

= 2 \cdot \tfrac{\pi}{2}

sin² averages ½ over [0, π].

= \pi

A circle of radius 1.

Your turn, guided

Your turn: inside $r = 3$ .

A = 9\pi

½ ∫₀^{2π} 9 dθ.

Practice ★

Use $A = \tfrac{1}{2}\displaystyle\int r^2\,d\theta$ to find the area inside $r = 9$ .
Use $A = \tfrac{1}{2}\displaystyle\int r^2\,d\theta$ to find the area inside $r = 4$ .
Use $A = \tfrac{1}{2}\displaystyle\int r^2\,d\theta$ to find the area inside $r = 6$ .

Watch out!

Practice ★★

Find the area of one petal of the rose $r = 3\cos(2\theta)$ (the petal for $-\tfrac{\pi}{4} \le \theta \le \tfrac{\pi}{4}$ ).

Find the area inside the cardioid $r = 5(1 + \cos\theta)$ .
Find the area inside the cardioid $r = 3(1 + \cos\theta)$ .

Exit ticket

Three questions to check you have got it.

Use $A = \tfrac{1}{2}\displaystyle\int r^2\,d\theta$ to find the area inside $r = 9$ .
Use $A = \tfrac{1}{2}\displaystyle\int r^2\,d\theta$ to find the area inside $r = 4$ .
Use $A = \tfrac{1}{2}\displaystyle\int r^2\,d\theta$ to find the area inside $r = 6$ .

Summary card

Polar area

$A = \tfrac{1}{2}\displaystyle\int_{\alpha}^{\beta} r^2\,d\theta$ .

Watch out: the ½ and the square.

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