This integral is the evaluation of the integrand e^x^3 on the area in the x,y plane bounded by
the x axis, the line x = 1 and the curve y = x^2 . Integrals like this can written as either
∫ dy ∫ dx or ∫ dx ∫ dy
with appropriate limits on the inner integral. The form ∫ dx ∫ dy is better in this case
because the integrand does not depend on y. The double integral is
∫ from 0 to 1 dx ∫ from 0 to x^{2} dy e^x^3
Evaluating the inner integral is trivial resulting in ∫ form 0 to 1 dx [ x^{2} e^x^3 ]
The x^{2} is an integrating factor, so the anti-derivative is just (1/3) e^x^3.
Thus the final result is (1/3) [ e -1)