Question

Find the area lying between the curve $y^2=4x$ and the line $y=2x$.

Answer 1

Given curves $C_1:y^2=4x,y=2x$ and $C_2:{4x}^2=4x$ intersects at $x=0and1$

Area bounded between the curves $C_1$ and $C_2$ is

$={\int }_0^1(C_1-C_2)dx$

$={\int }_0^1(2\sqrt{x}-2x)dx$

$={|2\frac{x^{\frac{3}{2}}}{\frac{3}{2}}-2\frac{x^2}{2}|}_0^1$

$={|\frac{4}{3}{x}^{\frac{3}{2}}-x^2|}_0^1$

$=\frac{4}{3}-1=\frac{1}{3}$