Chapter 11.3, Problem 1QQ

To determine

To determine: The correct answer for area of the region inside the polar curve $r=2\cos \theta$ and outside the polar curve $r=\cos \theta$ .

Answer to Problem 1QQ

The required correct option is (A).

Explanation of Solution

Given information:

The options:

(A) $3{\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}{\cos }^2\theta d\theta }$

(B) $3{\displaystyle \underset{0}{\overset{\pi }{\int }}{\cos }^2\theta d\theta }$

(C) $\frac{3}{2}{\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}{\cos }^2\theta d\theta }$

(D) $3{\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}\cos \theta d\theta }$

(E) $3{\displaystyle \underset{0}{\overset{\pi }{\int }}\cos \theta d\theta }$

Calculation:

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

It is known that in a polar coordinate system, if $r=f\left(\theta \right)$ is a curve; then the area of the region within the interval $\alpha \le \theta \le \beta$ is $A={\displaystyle {\int }_{\alpha }^{\beta }\frac{1}{2}{r}^{2}d\theta }={\displaystyle {\int }_{\alpha }^{\beta }\frac{1}{2}{\left(f\left(\theta \right)\right)}^{2}d\theta }$ .

The graph of the given curves is shown below:

  

It is seen that the shaded region between the curves lie within the interval $0\le \theta \le \frac{\pi }{2}$ .

Write the area of the given region as:

  $\begin{array}{c}A=\frac{1}{2}{\displaystyle {\int }_{\alpha }^{\beta }\left(r_2{}^2-r_1{}^2\right)d\theta }\\ =2\left({\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{1}{2}{\left(2\cos \theta \right)}^{2}d\theta }-{\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{1}{2}{\left(\cos \theta \right)}^{2}d\theta }\right)\\ =2{\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{1}{2}\left(4{\cos }^2\theta -{\cos }^2\theta \right)d\theta }\\ =3{\displaystyle {\int }_0^{\frac{\pi }{2}}{\cos }^2\theta d\theta }\end{array}$

Hence, the required correct option is (A).