What is the easiest way to evaluate this integral?

Question

\begin{equation*} \int_{0}^{64}\int_{\frac{1}{2}\sqrt[3]{y}}^{2} \frac{y^2}{\sqrt{x^{10} +1}} dxdy \end{equation*}

I'm probably doing something really wrong, because I'm stuck.

Any help will be appreciated.

Thanks.

Answer 1

Since the joint function is non-negative in the domain, and the domain equivalence is: $$(y\in [0,64]\cap x\in [\tfrac 12 \sqrt[3] y, 2]) \equiv (x\in [0,2]\cap y\in [0, 8 x^3])$$

Then we can change the order of the integration: $$\int_0^{64} y^2 \int_{(\sqrt[3] y / 2)}^{2} \frac{1}{\sqrt{x^{10}+1}} \operatorname d x \operatorname d y = \int_0^2\frac{1}{\sqrt{x^{10}+1}} \int_0^{8x^3} y^2 \operatorname d y \operatorname d x $$