integral from 2-0, integral from 4 to x^(3)/2, of x^(2)e^(y^(2)) dy dx
Because of that e^(y^2) you will want to reverse the order of integration. The curve y= (x^3)/2 and the line y= 4 will intersect at (2, 4) so y will go from 0 to 4. And, for each y, x will go from 0 to x= (2y)^{1/3}.
$\displaystyle \int_{y= 0}^4 \int_{x= 0}^{(2y)^{1/3}}x^2e^{y^2}dx dy$.