
iterated integral
Hello, I want to calculate:
$\displaystyle \int_{A} y d(x,y)$ where
$\displaystyle A={(x,y): y\ge0; x^2+y^2\le1; x^2+y^22x\le0 }$
So:
$\displaystyle \int_{A} y d(x,y) = \int^{1/2}_0 \int^{\sqrt{2xx^2}}_0 y dy dx + \int^{1}_{1/2} \int^{\sqrt{1x^2}}_0 y dy dx$
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.

1 Attachment(s)
I have made the graphic with A region, I think maybe there is a mistake with polar coordinate?
Thank you.