Hello, I want to calculate:

$\displaystyle \int_{A} y d(x,y)$ where

$\displaystyle A={(x,y): y\ge0; x^2+y^2\le1; x^2+y^2-2x\le0 }$

So:

$\displaystyle \int_{A} y d(x,y) = \int^{1/2}_0 \int^{\sqrt{2x-x^2}}_0 y dy dx + \int^{1}_{1/2} \int^{\sqrt{1-x^2}}_0 y dy dx$

Solving I have:

$\displaystyle \int_{A} y d(x,y) = \frac{5}{24}$

I have tried to solve it with polar coordinate:

$\displaystyle \int_{A} y d(x,y) = \int^{\pi/3}_0 \int^1_0 r^2 sin\theta dr d\theta + \int^{\pi/2}_{\pi/3} \int^{2cos\theta}_1 r^2 sin\theta dr d\theta$

Solving this one I have:

$\displaystyle \int_{A} y d(x,y) = \frac{1}{24}$

As we can see, second result is 5 times the first one, so there must be any mistakes.

Any suggestions?

Thank you very much.