$\displaystyle \int_{0}^{2}\int_{y^{2}}^{\sqrt[3]{y^{2}}}ye^{x^{2}}dxdy$

we solve it by changing the order of ontegration

at this integral

y goes from 0 till 2

x goes from $\displaystyle x=y^{2}$ till $\displaystyle x=\sqrt[3]{y^{2}}$

now we what first integrate by y then by x

so when we integrate by y: our intervals are $\displaystyle y=\sqrt{x}$ till $\displaystyle y=\sqrt[2]{y^{3}}$

what are the intervals when we integrate by x

?