Question

The area of the region above X-axis included between the parabola $y^2=x$ and the circle $x^2+y^2=2x$ in square units is

• A. $\frac{2}{3}-\frac{\pi }{4}$
• B. $\frac{\pi }{4}-\frac{3}{2}$
• C. $\frac{\pi }{4}-\frac{2}{3}$
• D. $\frac{3}{2}-\frac{\pi }{4}$

Answer 1

The correct option is C $\frac{\pi }{4}-\frac{2}{3}$


$y^2=x$....(1)
$x^2+y^2=2x$.......(2)
Equation (2) is a circle with centre (1, 0) and radius 1. Solving (1) and (2), we get the points of intersection (0, 0) and (1, 1)
Required area=$\underset{0}{\overset{1}{\int }}(\sqrt{1-(x-1{)}^2}-\sqrt{x})dx$
$={[\frac{x-1}{2}\sqrt{1-(x-1{)}^2}+\frac{1}{2}{\sin }^{-1}\frac{x-1}{1}-\frac{x^{3/2}}{\frac{3}{2}}]}_0^1$
$=-\frac{2}{3}+\frac{\pi }{4}$