Chapter 10.3, Problem 58E

To determine

To calculate: To solve the area of the region inside the polar curve.

Answer to Problem 58E

The area of the region is $8-\pi$ .

Explanation of Solution

Given information: The given curve as:

The area of the region inside the circle and outside the cardioid. The polar curves are $r=2$ and $r=2\left(1-\sin \theta \right)$ .

Calculation:

The sketch of the polar curves are:

  

The area of the given curve can be calculated as:

  $A=\frac{1}{2}{\displaystyle {\int }_a^b{r}^{2}d\theta }$ .

Based on the symmetry with respect to the $x-$ axis can be calculated as:

  $\begin{array}{c}A=2\pi -\frac{1}{2}{{\displaystyle {\int }_0^{\pi }\left[2\left(1-\sin \theta \right)\right]}}^{2}d\theta \\ A=2\pi -2{\displaystyle {\int }_0^{\pi }\left[\left(1-2\sin \theta +{\sin }^2\theta \right)\right]}d\theta \\ A=2\pi -2{\displaystyle {\int }_0^{\pi }\left[1-2\sin \theta +\frac{1-\cos 2\theta }{2}\right]}d\theta \\ A=2\pi -2{\left[\frac{3\theta }{2}+2\cos \theta -\frac{\sin 2\theta }{4}\right]}_0^{\pi }\\ A=2\pi -2\left[\frac{3\pi }{2}+2\cos \pi -\frac{\sin 2\pi }{4}-2\cos 0\right]\\ A=2\pi -3\pi +4+4\\ A=-\pi +8\\ A=8-\pi \end{array}$

The area of the curve is $8-\pi$ .