Write an integral that represents the area of the shaded region for $r = \cos 2 \theta \text { as }$ shown in the figure. Do not evaluate the integral.
A)$\int _ { - 5 \pi / 4 } ^ { - 3 \pi / 4 } ( \cos 2 \theta ) ^ { 2 } d \theta$
B)$\int _ { - 5 \pi / 2 } ^ { - 3 \pi / 2 } ( \cos 2 \theta ) ^ { 2 } d \theta$
C) $\frac { 1 } { 2 } \int _ { 3 \pi / 2 } ^ { 5 \pi / 2 } ( \cos 2 \theta ) ^ { 2 } d \theta$
D) $\frac { 1 } { 2 } \int _ { - 5 \pi / 4 } ^ { - 3 \pi / 4 } ( \cos 2 \theta ) ^ { 2 } d \theta$
E) $\frac { 1 } { 2 } \int _ { 3 \pi / 4 } ^ { 5 \pi / 4 } ( \cos 2 \theta ) ^ { 2 } d \theta$

Question

Write an integral that represents the area of the shaded region for $r = \cos 2 \theta \text { as }$ shown in the figure. Do not evaluate the integral. 
A)$\int _ { - 5 \pi / 4 } ^ { - 3 \pi / 4 } ( \cos 2 \theta ) ^ { 2 } d \theta$ 
B)$\int _ { - 5 \pi / 2 } ^ { - 3 \pi / 2 } ( \cos 2 \theta ) ^ { 2 } d \theta$ 
C) $\frac { 1 } { 2 } \int _ { 3 \pi / 2 } ^ { 5 \pi / 2 } ( \cos 2 \theta ) ^ { 2 } d \theta$ 
D) $\frac { 1 } { 2 } \int _ { - 5 \pi / 4 } ^ { - 3 \pi / 4 } ( \cos 2 \theta ) ^ { 2 } d \theta$ 
E) $\frac { 1 } { 2 } \int _ { 3 \pi / 4 } ^ { 5 \pi / 4 } ( \cos 2 \theta ) ^ { 2 } d \theta$

Quizplus · Accepted Answer

To find the shaded area, we need to integrate from where the curve intersects the x-axis (which is at $	heta = \frac{3\pi}{4}$ and $	heta = \frac{5\pi}{4}$) and then subtract the area under the curve for values of $	heta$ between those two points. Since the curve is symmetric about the line $	heta = \pi$, we can simplify the calculation by integrating over the interval $\left[\frac{3\pi}{4},\frac{5\pi}{4}ight]$ and then multiplying the result by $\frac{1}{2}$. Thus the integral we need is:

\begin{align*}
\frac{1}{2} \int_{3\pi/4}^{5\pi/4} (\cos 2	heta)^2\,d	heta
\end{align*}

which is option (E).

Write an Integral That Represents the Area of the Shaded r=cos⁡2θ as r = \cos 2 \theta \text { as }r=cos2θ as

Write an Integral That Represents the Area of the Shaded $r = \cos 2 \theta \text { as }$