Calculate the whole area encosed by the curve $y^2=x^4(a-x^2),a>0$

Question

Calculate the whole area encosed by the curve $y^2=x^4(a-x^2),a>0$.

I could not plot this curve,so could not find the area.I tried wolframalpha also.Here $a$ is not specified.Required area is $\frac{\pi a^2}{4}$.Please help me.

Answer 1

$\qquad\qquad$

The figure corresponds to the the case $a=1$.

The desired area is $${\frak A}=2\int_{-\sqrt a}^{\sqrt a}x^2\sqrt{a-x^2}dx$$ Now, the change of variables $ x=\sqrt{a}\cos(t/2)$ yields $${\frak A}=a^2\int_{ 0}^{2\pi}\cos^2\frac{t}{2}\sin^2\frac{t}{2}dt=\frac{a^2}{4} \int_{ 0}^{2\pi}\sin^2tdt=\frac{\pi a^2}{4}$$