Chapter 11.3, Problem 63E

To determine

To determine: The correct answer for the integral of the area of the region enclosed by the polar graph.

Answer to Problem 63E

The required correct option is (D).

Explanation of Solution

Given information:

The polar curve: $r=\sqrt{3+\cos \theta }$

The options:

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

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

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

(D) ${\displaystyle \underset{0}{\overset{\pi }{\int }}\left(3+\cos \theta \right)d\theta }$

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

Calculation:

The given curve is $r=\sqrt{3+\cos \theta }$ .

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 $0\le \theta \le 2\pi$ .

Write the area of the given region as:

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

Hence, the required correct option is (D).