Question

Sketch the region {(x, 0) : y = $\sqrt{4-x^2}$ and X - axis. Find the area of the region using integration.

Answer 1

Given region is $(x,0):y=\sqrt{4-x^2}$ and X - axis.

We have, $y=\sqrt{4-x^2}\Rightarrow y^2=4-x^2\Rightarrow x^2+y^2=4$

$\therefore$ Area of shaded region, $A=2{\int }_{-2}^2\sqrt{4-x^2}dx={\int }_{-2}^2\sqrt{2^2-x^2}dx{[\frac{x}{2}\sqrt{2^2-x^2}+\frac{2^2}{2}.si{n}^{-1}\frac{x}{2}]}_{-2}^2=\frac{2}{2}.0+2.\frac{\pi }{2}+\frac{2}{2}.0-2si{n}^{-1}(-1)=2.\frac{\pi }{2}+2.\frac{\pi }{2}=2\pi squnits$