<s>[INST] Evaluating $\iint_R\big(x^2+y^2\big)\,dA$ 
Evaluate the following double integral:
  $$\iint_R\big(x^2+y^2\big)\,dA,$$
  where $R$ is the region given by plane $x^2+y^2\leq a^2$.

My attempts:
\begin{align}
\iint_{R}\big(x^2+y^2\big)\,dA 
&=\int_{-a}^{a}\int_{-\sqrt{a^2-x^2}}^{\sqrt{a^2-x^2}}\big(x^2+y^2\big)\,dy\,dx\\ 
&=\int_{-a}^{a}\left(x^2y+\dfrac{y^3}{3}\right)_{-\sqrt{a^2-x^2}}^{\sqrt{a^2-x^2}}\,dx\\ 
&=\dfrac{2}{3}\int_{-a}^{a}\sqrt{a^2-x^2}\cdot\left(2x^2+a^2\right)dx.
\end{align}
I can't go further from here, please help.
 [/INST] Hint: Try $x=a\sin(t)$, but really polar coordinates would be the move here.
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