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How do I use Cylindrical Coordinates to find volume of a solid in the first Octant that is bounded by the cylinder $x^2 + y^2 = 2y$, the half cone $z = \sqrt{x^2 + y^2}$, and the $xy$-plane.

I have drawn the region of integration and obtained this:

$\int_0^2 \int_0^\sqrt{2y-y^2}\int_0^\sqrt{x^2 + y^2} dzdxdy$

Is this correct and from here were do I apply the cylindrical coordinates?

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I think I got it:

$\int_0^{\frac{\pi}{2}} \int_0^1\int_0^z r dzdrd\theta$

Can someone confirm if its correct.

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