<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] Whenever you have $\sqrt{a^2-x^2}$ in an integral, use the substitution $x=a \sin(\theta )$.
For information : 
In case of having  $\sqrt{x^2-a^2}$  use the substitution $x=a \sec(\theta )$.
In case of having  $\sqrt{x^2+a^2}$  use the substitution $x=a \tan(\theta )$
</s>