Question

Find the area of the region bounded by the curve $a{y}^2=x^3$, the $y-axis$ and the lines $y=a$ and $y=2a$.

Answer 1

Required area $={\int }_a^{2a}xdy$

$={\int }_a^{2a}{\left(a{y}^2\right)}^{1/3}dy$

$=a^{1/3}{\int }_a^{2a}{y}^{2/3}dy$

$=a^{1/3}{\int }_a^{2a}{\left[y^{2/3}\right]}_0^{2a}dy$

$=\frac{3}{5}{a}^{1/3}\{{\left(2a\right)}^{5/3}-a^{5/3}\}$