Q. 63 - Calculus

Question

Use a double integral to prove that the area of the circle with radius R and equation$r=2R\cos \theta is\pi R^2.$

Step-by-Step Solution

Answer

The area of the circle is

$A=\pi R^2$P

Step 1: Given information

The objective of this problem is to use double integral to prove that the area of the circle with radius R and equation $r=2R\cos \theta is\pi R^2.$

Step 1 : calculation

  Draw the circle

Plot of $r=2R\cos \theta$ 

Given circle is symmetrical about the horizontal axis. Therefore area of circle in polar form can be expressed as the twice of area of upper half circle.

$A=2{\int }_6^{a_1}{\int }_5^{x}rdrd\theta$

Here, ${\theta }_1=0,{\theta }_2=\frac{\pi }{2}and{r}_1=0,r_2=r$

$A=2{\int }_0^{\pi /2}{\int }_0^{r-2\pi \cos \theta }rdrd\theta$

Integrate with respect to r first

$A=2{\int }_0^{*/2}{\left[\frac{r^2}{2}\right]}_0^{2\pi ene}d\theta \left[\int x^{n}dx=\frac{x^{n+1}}{n+1}+C\right]$

$A=2{\int }_0^{\pi /2}\left[\frac{(2R\cos \theta {)}^2-0}{2}\right]$

$$\begin{gathered} A=2R^2{\int }_0^{\pi /2}\left[2{\cos }^2\theta \right]d\theta \\ A=2R^2{\int }_0^{\pi /2}[1+\cos 2\theta ]d\theta \end{gathered}$$

Integrate with respect to$\theta$

$$\begin{gathered} A=2R^2{\left[\theta +\frac{1}{2}\sin 2\theta \right]}_0^{x/2}\left[\int \cos xdx=\sin x+C\right] \\ A=2R^2\left[\frac{\pi }{2}+\frac{1}{2}\sin \pi -0\right] \end{gathered}$$

$A=\pi R^2$