Find the area of the region specified in polar coordinates.
-the region enclosed by the curve $r = 6 \cos \theta$
A) $9 \pi$
B) $12 \pi$
C) $18 \pi$
D) $36 \pi$

Question

Quizplus · Accepted Answer

The area $A$ enclosed by a polar curve $r = f(\theta)$ from $\theta = a$ to $\theta = b$ is given by the integral $A = \frac{1}{2} \int_{a}^{b} r^2 d\theta$. For $r = 6 \cos \theta$, the curve is a circle in the polar coordinate system, and it is symmetric about the vertical axis. The curve intersects the polar axis ($\theta = 0$) at $r = 6$ and again at $\theta = \pi$, indicating the full circle is described as $\theta$ goes from $0$ to $\pi$. Thus, the area is $A = \frac{1}{2} \int_{0}^{\pi} (6 \cos \theta)^2 d\theta = 18 \pi$, which corresponds to choice A.

Find the Area of the Region Specified in Polar Coordinates r=6cos⁡θr = 6 \cos \thetar=6cosθ

Find the Area of the Region Specified in Polar Coordinates $r = 6 \cos \theta$