Find the area of the specified region.
-Shared by the cardioids $r = 4 ( 1 + \sin \theta )$ and $r = 4 ( 1 - \sin \theta )$
A) 1616
B) $8 ( 3 \pi - 8 )$
C) $8 ( 3 \pi + 8 )$
D) $4 ( 5 \pi + 8 )$

Question

Quizplus · Accepted Answer

We can see from the given equations that the cardioids intersect at the origin and at the points where $\sin \theta = \frac{1}{2}$, which occur at $\theta = \frac{\pi}{6}$ and $\theta = \frac{5 \pi}{6}$.

To find the area of the region enclosed by the two cardioids, we can integrate the formula for the area of a polar region:

$$\int_{\alpha}^{\beta} \frac{1}{2} r(\theta)^2 d\theta$$

where $\alpha$ and $\beta$ are the angles at which the cardioids intersect.

Thus, the area of the region is:

\begin{align*}
\int_{\pi/6}^{5\pi/6} \frac{1}{2} \left[(4(1+\sin\theta))^2 - (4(1-\sin\theta))^2\right] d\theta &= \int_{\pi/6}^{5\pi/6} \frac{1}{2} \left[32\sin\theta\right] d\theta \
&=16\int_{\pi/6}^{5\pi/6} \sin\theta\ d\theta \
&=16\left[-\cos\theta\right]_{\pi/6}^{5\pi/6} \
&=16\left[-\left(-\frac{1}{2}\right)+\frac{1}{2}\right] \
&=8
\end{align*}

Therefore, the area of the specified region is 8, which corresponds to choice B.

Find the Area of the Specified Region r=4(1+sin⁡θ)r = 4 ( 1 + \sin \theta )r=4(1+sinθ)

Find the Area of the Specified Region $r = 4 ( 1 + \sin \theta )$