Step 1:

$2ay^2=x(x-a)^2$ is resolved about the x-axis.

$a > 0$ The curve is symmetric about x axes, passes through the orgin, does not exist for $x < 0$ and a loop is formed between $x=0\; and\; x=a$

Step 2:

Volume formed by rotating the loop

$V= \pi \int \limits_0^a y^2 dx$

$\quad=\large\frac{\pi}{2a} $$\int \limits_0^a x(x-a) ^2 dx$

$\quad= \large\frac{\pi}{2a} $$\int \limits_0^a (x^3-2ax^2 +a^2 x)dx$

$\quad= \large\frac{\pi}{2a} \bigg[\frac{x^4}{4}-\frac{2ax^3}{3}+\frac{a^2x^2}{2}\bigg]_0^a$

$\quad= \large\frac{\pi}{2a} \bigg[\frac{a^4}{4}-\frac{2ax^4}{3}+\frac{a^4}{2}\bigg]$

$\quad= \large\frac{\pi}{24a}$$ \bigg[3a^4-8a^4+6a^4\bigg]$

$\quad=\large\frac{\pi a^3}{24} $$units$