Question

Use a double integral with polar coordinates to prove that the area of a sector with central angle $\varphi$ in a circle of radius R is given by $A=\frac{1}{2}\varphi R^2.$

Short answer

The area of a sector with central angle $\varphi$is$A=\frac{1}{2}\varphi R^2$

Step-by-step solution

Given information

The objective of this problem is to use double integral to prove that the area of a sector with central angle $\varphi is\frac{1}{2}\varphi R^2.$

calculation

In Cartesian system the equation of a circle vith radius R centered at origin is

$x^2+y^2=R^2$

Area of sector in double integration can be expressed as

$A=\iint dxdy$

In polar form

A=\int_{\phi}^{\phi} \int_{n}^{n} r d r d \theta$A=\iint dxdy$

Here, $\varphi =0,{\varphi }_2=\varphi and{r}_1=0,r_2=R$

Then

$$\begin{gathered} A={\int }_0^{\varphi }{\int }_0^{R}rdrd\theta \\ A={\int }_0^{\varphi }\left[{\int }_0^{R}rdr\right]d\theta A={\int }_0^{\varphi }{\left[\frac{r^2}{2}\right]}_0^{R}d\theta \end{gathered}$$

$A={\int }_0^{\varphi }\left[\frac{R^2-0}{2}\right]d\theta$

$A=\frac{1}{2}{R}^2{\int }_0^{\varphi }d\theta$

$A=\frac{1}{2}{R}^2[\theta {]}_0^{\varphi }$

$A=\frac{1}{2}\varphi R^2$

$A=\frac{1}{2}\varphi R^2$