Question

Find the area bounded by the curve $y^2=4ax$, $X$-axis and the lines $x=0$ and $x=a$.

Answer 1

Given, $y^2=4ax$

Required area $\left(A\right)={\int }_0^{a}ydx$

$={\int }_0^{a}2\sqrt{ax}dx$

$=2\sqrt{a}{\left[\frac{x^{3/2}}{\frac{3}{2}}\right]}_0^a$

$=2\sqrt{a}\cdot \frac{2}{3}[x^{3/2}{]}_0^a$

$=\frac{4}{3}\sqrt{a}(a^{3/2}-0)$

$=\frac{4}{3}{a}^{\frac{1}{2}+\frac{3}{2}}$

$=\frac{4}{3}{a}^2$ sq. units