Question

Area lying between the curve $y^2=4x$ and $y=2x$ is

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

Answer 1

The correct option is B $\frac{1}{3}$

The area lying between the curve, $y^2=4x$ and $y=2x$ is represented by the shaded area $OBAO$ as shaded in the diagram.
The points of intersection of these curves are $O(0,0)$ and $A(1,2)$.
We draw $AC$ perpendicular to $x$-axis such that the coordinates of $C$ are $(1,0)$.
$\therefore$ Area $OBAO$ $=Area\left(\Delta OCA\right)-Area\left(OCABO\right)$
$={\int }_0^{1}2xdx-{\int }_0^{1}2\sqrt{x}dx$
$=2{\left[\frac{x^2}{2}\right]}_0^1-2{\left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]}_0^1$
$=|1-\frac{4}{3}|$
$=|-\frac{1}{3}|=\frac{1}{3}$ sq. units
Thus, the correct answer is B.