Question

Solve ${\int }_0^a\frac{\sqrt{a^2{b}^2-b^2{x}^2}}{a}dx$

Answer 1

${\int }_0^a\frac{\sqrt{a^2{b}^2-b^2{x}^2}}{a}dx$

$=\frac{b}{a}{\int }_0^a\sqrt{a^2-x^2}dx$

$=\frac{b}{a}{[\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}{\sin }^{-1}(\frac{x}{a}\left)\right]}_0^a$

$=\frac{b}{a}\left[\right(\frac{a}{2}\times 0+\frac{a^2}{2}{\sin }^{-1}1)-0]$

$=\frac{b}{a}[\frac{a^2}{2}\times \frac{\pi }{2}]$

$=\frac{ab\pi }{4}$ (Ans)