Find the area of inside $r = 88 \cos \theta \text { and outside } r = 44 \text { by sketching the graph of the }$ equations using the graphing utility.
A) $\frac { 1936 \pi } { 3 } - \frac { 968 \sqrt { 2 } } { 1 }$
B) $- \frac { 1936 \pi } { 3 } - \frac { 968 \sqrt { 2 } } { 1 }$
C) $\frac { 1936 \pi } { 3 } + \frac { 968 \sqrt { 3 } } { 1 }$
D) $\frac { 968 \pi } { 1 } + \frac { 1936 \sqrt { 3 } } { 3 }$
E) $\frac { 968 \pi } { 1 } - \frac { 1936 \sqrt { 3 } } { 3 }$

Question

Find the area of inside $r = 88 \cos \theta \text { and outside } r = 44 \text { by sketching the graph of the }$ equations using the graphing utility. 
A) $\frac { 1936 \pi } { 3 } - \frac { 968 \sqrt { 2 } } { 1 }$
B) $- \frac { 1936 \pi } { 3 } - \frac { 968 \sqrt { 2 } } { 1 }$
C) $\frac { 1936 \pi } { 3 } + \frac { 968 \sqrt { 3 } } { 1 }$
D) $\frac { 968 \pi } { 1 } + \frac { 1936 \sqrt { 3 } } { 3 }$
E) $\frac { 968 \pi } { 1 } - \frac { 1936 \sqrt { 3 } } { 3 }$

Quizplus · Accepted Answer

After plotting the two equations in a graphing utility, we can see that the region enclosed by $r=88\cos	heta$ is a cardioid, while the region outside of $r=44$ is just the whole plane with a circle of radius 44 removed. To find the area of the shaded region, we need to set up an integral in polar coordinates. Since we want the area that is inside $r=88\cos	heta$ and outside $r=44$, we need to subtract the area of the circle of radius 44 from the area of the cardioid. The integral that gives the area of the cardioid is:
\begin{align*}
\frac{1}{2}\int_{-\pi/2}^{\pi/2}(88\cos	heta)^2\ d	heta&=\frac{1}{2}\int_{-\pi/2}^{\pi/2}7744\cos^2	heta\ d	heta\
&=\frac{1936}{3}\cdot 2\cdot 2\
&=\frac{1936}{3}\cdot 4.
\end{align*}
The area of the circle of radius 44 is $\pi(44^2)$. Therefore, the area of the shaded region is:
\begin{align*}
\frac{1936}{3}\cdot 4-\pi(44^2)&=\frac{1936}{3}\cdot\frac{4}{1}-\pi\cdot (44\cdot 44)\
&=\frac{1936\cdot 4}{3}-1936\pi\
&=1936\left(\frac{4}{3}-\piight)\
&\approx 968\cdot 0.2324\
&\approx 224.759.
\end{align*}
Therefore, the answer is $\boxed{	extbf{(C)}}$.

Find the Area of Inside r=88cos⁡θ and outside r=44 by sketching the graph of the r = 88 \cos \theta \text { and outside } r = 44 \text { by sketching the graph of the }r=88cosθ and outside r=44 by sketching the graph of the

Find the Area of Inside $r = 88 \cos \theta \text { and outside } r = 44 \text { by sketching the graph of the }$