Unsolvable iterated integral: $\int_0^8\int_{y^{1/3}}^2 4e^{x^4} dx\,dy$

Question

Having issues solving the following iterated integral, which appears to not have a tangible antiderivative. $$\int_0^8\int_{y^{1/3}}^2 4e^{x^4} dx\,dy$$

So, I figure it as a graph with the domain of $(x,y): 0≤y≤8, y{^{\frac{1}{3}}≤x≤2}$, making the graph bound by the lines $y=x^3, x=2$, and $y=8$. However, switching the integrals around does not clarify this equation whatsoever. Should I take the ln of the entire function? Please help.

Answer 1

HINT:

$$\int_0^8\int_{y^{\frac{1}{3}}}^2 4e^{x^4} \,dx\,dy=\int_0^2 4e^{x^4}\int_0^{x^3} \,dy\,dx$$