Chapter 10.3, Problem 64E

To determine

To check: The correct option from the given options.

Answer to Problem 64E

The correct option is (D).

Explanation of Solution

Given information: The given polar curve is $r=4\cos \left(3\theta \right)$ .

Formula used: The formula to find the region between the origin and curve $r=f\left(\theta \right)$ from $\left[a,b\right]$ is given by.

  $A={\displaystyle \underset{a}{\overset{b}{\int }}\frac{1}{2}{r}^{2}d\theta }$

Calculation:

Plot the graph of polar curve $r=4\cos \left(3\theta \right)$ on the plane. $\theta =\pi$

  

Figure (1)

The one petal of the rose lies between the interval $\theta =-\frac{\pi }{6}$ and $\theta =\frac{\pi }{6}$ .

To find the area of the region inside the circle, integrate the function from $\theta =-\frac{\pi }{6}$ and $\theta =\frac{\pi }{6}$ .

  $\begin{array}{c}A=\frac{1}{2}{\displaystyle \underset{-\frac{\pi }{6}}{\overset{\frac{\pi }{6}}{\int }}{\left(4\cos 3\theta \right)}^{2}d\theta }\\ =\frac{16}{2}{\displaystyle \underset{-\frac{\pi }{6}}{\overset{\frac{\pi }{6}}{\int }}{\cos }^{2}3\theta d\theta }\\ =8{\displaystyle \underset{-\frac{\pi }{6}}{\overset{\frac{\pi }{6}}{\int }}{\cos }^{2}3\theta d\theta }\end{array}$

Thus, the correct option is (D).