Question

find the area of the shaded region.
$r^{2}=sin(5\theta)$

Explanation:

Step1: Recall area formula in polar coordinates

The area $A$ of a polar - curve $r = f(\theta)$ is given by $A=\frac{1}{2}\int_{\alpha}^{\beta}r^{2}d\theta$. For the curve $r^{2}=\sin(5\theta)$, we need to find the limits of integration for one petal. We set $\sin(5\theta)=0$, then $5\theta = k\pi$, $\theta=\frac{k\pi}{5}$, $k\in\mathbb{Z}$. For one petal, we can take $\alpha = 0$ and $\beta=\frac{\pi}{5}$.

Step2: Calculate the area of one petal

Substitute $r^{2}=\sin(5\theta)$ into the area formula:
\[

$$\begin{align*} A&=\frac{1}{2}\int_{0}^{\frac{\pi}{5}}\sin(5\theta)d\theta\\ \end{align*}$$

\]
Let $u = 5\theta$, then $du=5d\theta$ and $d\theta=\frac{1}{5}du$. When $\theta = 0$, $u = 0$; when $\theta=\frac{\pi}{5}$, $u=\pi$.
\[

$$\begin{align*} A&=\frac{1}{2}\times\frac{1}{5}\int_{0}^{\pi}\sin(u)du\\ &=\frac{1}{10}[-\cos(u)]_{0}^{\pi}\\ &=\frac{1}{10}[-\cos(\pi)+\cos(0)]\\ &=\frac{1}{10}(1 + 1)\\ &=\frac{1}{5} \end{align*}$$

\]

Answer:

$\frac{1}{5}$