Evaluate the integral ∫ from 0 to 1 ∫ from sqrt(y) to 1 of e^x^3 dx dy by reversing the order of integration.

Please show all your work.

Evaluate the integral ∫ from 0 to 1 ∫ from sqrt(y) to 1 of e^x^3 dx dy by reversing the order of integration.

Please show all your work.

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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)

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