Question

Prove that the area enclosed by the curve with equation $x^2+y^2=r^2$, while taking the horizontal strips parallel to x-axis is $\pi r^2$ sq. units.

Answer 1

Equation $x^2+y^2=r^2$, represents a circle with centre $(0,0)$.
$\Rightarrow x=\sqrt{r^2-y^2}$
Area to be calculated can be simplified as:
Area of whole circle= 4(area of the region $ABOA$)
Required area $=4\underset{0}{\overset{r}{\int }}\sqrt{r^2-y^2}dy$
$=4{[\frac{y}{2}\sqrt{r^2-y^2}+\frac{r^2}{2}{\sin }^{-1}\left(\frac{y}{r}\right)]}_0^r$
$=4[0+\frac{r^2}{2}{\sin }^{-1}(1)-0]$
$=\pi r^2\text{sq. units.}$

Hence proved.