Question

Area of the region bounded by the curve $y=2\sqrt{1-x^2}$ and the $x$-axis, is

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

Answer 1

The correct option is A $\frac{\pi }{2}$

$y=2\sqrt{1-x^2}$

$\Rightarrow x=\sqrt{1-\frac{y^2}{4}}$

$A={\int }_0^2\sqrt{1-\frac{y^2}{4}}dy$

$y=2\sin u$

$=\int 2{\cos }^{2}udu$

$=2\int \frac{1+\cos 2u}{2}du$

$=\int 1+\cos 2udu$

$={\sin }^{-1}\left(\frac{1}{2}y\right)+\frac{1}{2}\sin \left(2{\sin }^{-1}\left(\frac{1}{2}y\right)\right)+C$

$A={\int }_0^2\sqrt{1-\frac{y^2}{4}}dy=\frac{\pi }{2}-0=\frac{\pi }{2}$