Chapter 10.3, Problem 48E

To determine

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

Answer to Problem 48E

The area of the region is $4$ .

Explanation of Solution

Given information: The given curve as:

The area of the region inside the six-petaled rose $r^2=2\sin \left(3\theta \right)$ .

Calculation:

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:

  $\begin{array}{c}A=6\times \frac{1}{2}{\displaystyle {\int }_0^{\frac{\pi }{3}}\left(2\sin \left(3\theta \right)\right)}d\theta \\ A=-6{\left(\frac{\cos 3\theta }{3}\right)}_0^{\frac{\pi }{3}}\\ A=-2\left[\cos \pi -\cos 0\right]\\ A=-2\left[-1-1\right]\\ A=4\end{array}$

The area of the curve is $4$ .