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\(\displaystyle 0=x^4\left(a^2-x^2 \right)=x^4(a+x)(a-x)\)

And so we find these intercepts are at:

\(\displaystyle x=0,\,a\)

And so we may state:

\(\displaystyle A(a)=\frac{4}{a^2}\int_0^a x^2\sqrt{a^2-x^2}\,dx\)

At this point, we may consider the substitution:

\(\displaystyle x=a\sin(\theta)\,\therefore\,dx=a\cos(\theta)\)

So, I will now stop at this point to give you a chance to take it from here.