Chapter 10.3, Problem 63E

To determine

To check: The correct option from the given options.

Answer to Problem 63E

The correct option is (D).

Explanation of Solution

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

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 $r=\sqrt{3+\cos \theta }$ on the plane. $\theta =\pi$

  

Figure (1)

From the figure, it can be observed that the curve is symmetric about the $x$ -axis. So, the area of the region inside the curve is double of the area above the $x$ -axis.

To find the area of the region inside the circle, integrate the function from $0$ to $\pi$ .

  $\begin{array}{c}A=2\cdot {\displaystyle \underset{0}{\overset{\pi }{\int }}\frac{1}{2}{\left(\sqrt{3+\cos \theta }\right)}^{2}d\theta }\\ ={\displaystyle \underset{0}{\overset{\pi }{\int }}{\left(\sqrt{3+\cos \theta }\right)}^{2}d\theta }\\ ={\displaystyle \underset{0}{\overset{\pi }{\int }}\left(3+\cos \theta \right)d\theta }\end{array}$

Thus, the correct option is (D).