Double Integrals

Question

Evaluate the double integral over the given region R.

∫∫ xye^{xy^2} dA R: 0≤x≤2, 0≤y≤1

Answer 1

∫₀²∫₀¹ x y e^{xy^2} dy dx

 

This problem is easier to integrate in y first, since you can do a substitution of u=y², du = 2y dy, 1/2 du = y dy to get

 

1/2 ∫₀²∫₀¹ x e^xu du dx

 

Treat the x's as constants, and you get

= 1/2 ∫₀² x [1/x e^xu]₀¹ dx

= 1/2 ∫₀² e^x-1 dx

 

This is now an integral you should be able to do easily. If you have futher questions, please comment!