Evaluate the following iterated integral in cartesian variables ∫ ∫ xy(√x^2+y^2) dydx. The bounds are y=1, y=0, x=1, x=0

Answer 1

The answer to the question is 2/15 (2√2-1). The complete solution is difficult to type using this message, so here is a quick outline of the steps:

1. To perform the inner integral (in y), use the substitution u = y². Then du = 2y dy, and the integrand becomes x √(x²+u) du/2. This will give two integrals in the x variable, one of which has integrand x(x²+1)^3/2 and the other is just x⁴. (These integrals are multiplied by constants you need to keep track of.)
2. To perform the first x-integral, use the substitution v = x²+1 and proceed as with the previous step. Note that the limits for v are from 1 to 2 as x ranges from 0 to 1.

Once the integrals are performed and evaluated, easy simplification gives the answer above.