# Double integral with substitution

1. May 2, 2012

### carlosbgois

1. The problem statement, all variables and given/known data
Evaluate (using a substitution) $\int\int_{B}x^{2}+2y dxdy$ where $B=\{(x, y) | x^{2}+y^{2}≤4\}$

3. 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

2. May 2, 2012

### 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?

3. May 2, 2012

### 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

4. May 2, 2012

### carlosbgois

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