$$\Large \int\limits_{0}^{8}\int\limits_{0}^{y^{1/3}}\sqrt{x^4+1}dxdy$$

Question

geerky42 · Answer

I changed the order of integration, then when I tried to evaluate it, I am stuck at $\Large \displaystyle \int_{0}^2 8\sqrt{x^4+1} - x^3\sqrt{x^4+1}  dx$
How can I solve that?

geerky42 · Answer

Any idea? @SithsAndGiggles

anonymous · Answer

Not for the first term. There's doesn't seem to be a closed form for it. The second term could be integrated with a substitution though.

anonymous · Answer

Have you tried converting to polar coordinates?

geerky42 · Answer

No, I haven't because I don't see how that would helps me.

geerky42 · Answer

Maybe I made mistake when I switch integral order, but I checked and rechecked, nothing is wrong. 

But it seem like it cannot be solved by hand.

anonymous · Answer

No, your work is right. And you're right about not being able to solve it by hand, though you can try to approximate it with the first few terms of a the power series for $\sqrt{x^4+1}$.

I don't think there's any easy way to integrate what you have so far.

And you can ignore the polar-conversion suggestion, it only serves to make things I harder from the looks of it.

geerky42 · Answer

hmm, ok. thanks for helps though.