Find the area of the region specified in polar coordinates.
-the region enclosed by the curve r = 6 cos $\theta$
A) 18 $\pi$
B) 9 $\pi$
C) 12 $\pi$
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 the curve $r = 6\cos(\theta)$, the curve is symmetric about the vertical axis, and it completes one full petal as $\theta$ goes from $0$ to $\pi$. Thus, we integrate from $0$ to $\pi$:$$A = \frac{1}{2} \int_{0}^{\pi} (6\cos(\theta))^2 d\theta = 18 \int_{0}^{\pi} \cos^2(\theta) d\theta$$Using the double angle formula, $\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$, the integral becomes:$$A = 18 \int_{0}^{\pi} \frac{1 + \cos(2\theta)}{2} d\theta = 9[\theta + \frac{1}{2}\sin(2\theta)]_{0}^{\pi} = 9\pi$$Thus, the area of the region enclosed by the curve is $9\pi$.

Find the Area of the Region Specified in Polar Coordinates θ\thetaθ

Find the Area of the Region Specified in Polar Coordinates $\theta$