Evaluate the integral?

Question

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.

Answer 1

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)