Question

${\int }_0^a\sqrt{a^2-x^2}dx$ is equal to

• A. $\pi a^2$
• B. $\frac{1}{2}\pi a^2$
• C. $\frac{1}{3}\pi a^2$
• D. $\frac{1}{4}\pi a^2$

Answer 1

Answer: (D)

${\int }_0^a\sqrt{a^2-x^2}dx$
$={\left[\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}{\sin }^{-1}\left(\frac{x}{a}\right)\right]}_0^a$
$=\left[0+\frac{a^2}{2}{\sin }^{-1}(1)-0-\frac{a^2}{2}{\sin }^{-1}(0)\right]$
$=\frac{a^2}{2}.\frac{\pi }{2}-0=\frac{a^2\pi }{4}$