Question

Determine the area under the curve$y=\sqrt{a^2-x^2}$ included between the lines x = 0 and x = a.

Answer 1

Given equation of the curve is $y=\sqrt{a^2-x^2}$

$\Rightarrow y^2=a^2-x^2\Rightarrow y^2+x^2=a^2$

$\therefore$ Required area of shaded region, $A={\int }_0^a\sqrt{a^2-x^2}dx={[\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}si{n}^{-1}\frac{x}{a}]}_0^a=[0+\frac{a^2}{2}si{n}^{-1}(1)-0-\frac{a^2}{2}si{n}^{-1}0]=\frac{a^2}{2}.\frac{\pi }{2}=\frac{\pi a^2}{4}squnits$