Question

$\underset{0}{\overset{1}{\int }}\sqrt{x\left(1-x\right)}dx$ equals

(a) π/2
(b) π/4
(c) π/6
(d) π/8

Answer 1

(d) $\mathrm{\pi }$/8

$$\begin{gathered} \mathrm{Let},I={\int }_0^1\sqrt{x\left(1-x\right)}dx \\ ={\int }_0^1\sqrt{x-x^2}dx \\ ={\int }_0^1\sqrt{\frac{1}{4}-\left(x^2-x+\frac{1}{4}\right)}dx \\ ={\int }_0^1\sqrt{{\left(\frac{1}{2}\right)}^2-{\left(x-\frac{1}{2}\right)}^2}dx \\ ={\left[\frac{\left(x-\frac{1}{2}\right)}{2}\sqrt{x-x^2}+\frac{1}{2}\times \frac{1}{4}{\sin }^{-1}\left(2x-1\right)\right]}_0^1 \\ =\frac{1}{8}{\left[{\sin }^{-1}\left(1\right)-{\sin }^{-1}\left(-1\right)\right]}_0^1 \\ =\frac{1}{8}\left[\frac{\mathrm{\pi }}{2}+\frac{\mathrm{\pi }}{2}\right] \\ =\frac{\mathrm{\pi }}{8} \end{gathered}$$