Help with double integral!?

Question

I need to solve the double integral of:

y * sqrt(1 + x + y) dxdy , where both limits of integration are 0 to 1

I cannot figure out how to do this by hand! I CANNOT use a calculator or computer program or anything like that.

Answer 1

The x-integration is easy. If you need a substitution, try
u = 1+x+y; then du = dx. The result after the x-integration is

y*(2/3)(1 + x + y)^(3/2) evaluated at x=0 and x=1
= (2/3)y [(2+y)^(3/2) - (1+y)^(3/2)]

Then, to integrate y (2+y)^(3/2), let u = 2+y; then y = u-2, dy = du and you get

integral of (u - 2) u^(3/2) du
= integral of (u^(5/2) - 2 u^(3/2)) du

and similarly for integrating y(1+y)^(3/2)