How do I take the double integral e^(y/x) dx dy demarcated by the region D={(x,y):x^2≤y≤x^3≤8}?

See if you can rewrite D, so the bounds on y are in terms of x, and the bounds of x are in terms of y (or vice versa).

You can then use Fubini's theorem to first integrate w.r.t. y and then w.r.t. x over the bounds.

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