Question

Evaluate: $\stackrel{b}{\underset{a}{\int }}\sqrt{(x-a)(b-x)}dx$.

Answer 1

${\int }_a^b\sqrt{(x-a)(b-x)}dx$

$={\int }_a^b\sqrt{bx-x^2-ab+ax}dx$

$={\int }_a^b\sqrt{(a+b)x-ab-x^2}dx$

$={\int }_a^b\sqrt{\frac{a^2+b^2+2ab}{4}-ab-{(x-\frac{(a+b)}{2})}^2}$

$={\int }_a^b\sqrt{{\left(\frac{a-b}{2}\right)}^2-{(x-\frac{(a+b)}{2})}^2}$

$=\frac{(x-\frac{a+b}{2})\sqrt{{\left(\frac{a-b}{2}\right)}^2-{(x-\frac{(a+b)}{2})}^2}}{2}+\frac{{(a-b)}^2}{8}{\sin }^{-1}\left(\frac{(x-\frac{(a+b)}{2})}{\left(\frac{a-b}{2}\right)}\right)+C$

$=\frac{(2x-a-b)\sqrt{(x-a)(b-x)}}{4}+\frac{{(a-b)}^2}{8}{\sin }^{-1}\left(\frac{2x-a-b}{a-b}\right)+C$

Hence,

${\int }_a^b\sqrt{(x-a)(b-x)}dx=\frac{(2x-a-b)\sqrt{(x-a)(b-x)}}{4}+\frac{{(a-b)}^2}{8}{\sin }^{-1}\left(\frac{2x-a-b}{a-b}\right)+C$