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Evaluate the integral?

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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Richard P. | Fairfax County Tutor for HS Math and ScienceFairfax County Tutor for HS Math and Sci...
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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 x2  dy e^x^3 
  Evaluating the inner integral is trivial  resulting in ∫ form 0 to 1  dx [ x2 e^x^3 ]
  The x2 is an integrating factor, so the anti-derivative is   just  (1/3) e^x^3.
  Thus the final result is  (1/3) [ e -1)