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