Answer to Question #180261 in Calculus for Prathamesh

Question

Find the area of the circle x2 + y2 = r2.

Answer 1

Given equation of circle is:

$x^2+y^2=r^2$

$\Rightarrow y=\sqrt{r^2-x^2}$

Area of circle =4 $\times$ Area of first quadrant

        $=4 \int_0^rydx$

 

        $=4\int_0^r\sqrt{r^2-x^2}dx$

        $=4[\dfrac{x}{2}\sqrt{r^2-x^2}+\dfrac{r^2}{2}sin^{-1}\dfrac{x}{r}]_0^r$

        $=4[\dfrac{r^2}{2}sin^{-1}-0]$

        

$=4\times \dfrac{r^2\pi}{4}$

        $= \pi r^2 \text{ square units }$