Double integral with substitution

[carlosbgois]

Homework Statement

Evaluate (using a substitution) \int\int_{B}x^{2}+2y dxdy where B=\{(x, y) | x^{2}+y^{2}≤4\}

The Attempt at a Solution

I attempted a solution using polar coordinates, so the integral becomes \int\int_{B_{ρθ}}(ρ^{2}cos^{2}(θ)+2ρsin(θ)) ρ dρdθ, and the integration intervals are 0≤ρ≤2, 0≤θ≤\pi. Solving it using Fubini's thorem my result was \frac{32}{3}, but the solution given by the book is 4\pi.

Where did I go wrong?
Thanks

 

[Dick]

Your polar coordinate setup is right, except you want the upper limit to be 2pi. But the answer is completely wrong. How did that happen? Can you show us?

 

[carlosbgois]

Indeed it is 2pi, my mistake. I'll attempt a solution again, and if I don't get it i'll show it
Thank you

 

[carlosbgois]

Done, I did forget the square in the cosine hehe, many thanks