Find the area of the region specified in polar coordinates.
-the region enclosed by the curve $r = 2 + \cos \theta$
A) $5 \pi$
B) $\frac { 9 } { 2 } \pi$
C) $\frac { 5 } { 2 } \pi$
D) $3 \pi$

Question

Quizplus · Accepted Answer

The area enclosed by a polar curve $ r = f(\theta) $ from $\theta = a$ to $\theta = b$ is given by $\frac{1}{2} \int_{a}^{b} (f(\theta))^2 \, d\theta$. For $ r = 2 + \cos \theta $, the curve is a limaçon with an inner loop. The area is calculated over $[0, 2\pi]$, resulting in $\frac{1}{2} \int_{0}^{2\pi} (2 + \cos \theta)^2 \, d\theta = \frac{9}{2} \pi$.

Find the Area of the Region Specified in Polar Coordinates r=2+cos⁡θr = 2 + \cos \thetar=2+cosθ

Find the Area of the Region Specified in Polar Coordinates $r = 2 + \cos \theta$