Find the area inside the polar curve $r=2-2 \sin \theta$.
A) $6 \pi$
B) $3 \pi$
C) $12 \pi$
D) $9 \pi$

Question

Quizplus · Accepted Answer

The area inside the polar curve is given by the integral of $r^2/2$ from $\theta=0$ to $\theta=\pi$. So, the area is$$\int_0^\pi \frac{(2-2\sin \theta)^2}{2} d\theta = \int_0^\pi (2-4\sin \theta + 4\sin^2 \theta) d\theta$$$$= \int_0^\pi (2-4\sin \theta + 2-2\sin 2\theta) d\theta$$$$= \int_0^\pi 4 - 4\sin \theta - 2\sin 2\theta d\theta$$$$= 4\theta + 4\cos \theta + \cos 2\theta \bigg|_0^\pi$$$$= 4\pi - 4 + 1 = 6 \pi$$

Find the Area Inside the Polar Curve r=2−2sin⁡θr=2-2 \sin \thetar=2−2sinθ A)

Find the Area Inside the Polar Curve $r=2-2 \sin \theta$
A)