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We need to find the volume of the solid bounded by the $xy$-plane, the cylinder $x^2+y^2=2x$, and the cone $z=\sqrt{x^2+y^2}.$ The volume is given by the triple integral $\int \int \int_S \,dx\,dy\,dz,$where S is the region of the bounded solid.

I am trying to use spherical co-ordinates where the integral becomes $\int \int \int_T r^2 \phi \,dr\,d\theta\,d\phi$ but I am unable to set the correct bounds for the $r,\theta, \phi$ integrals. The order of integration I am trying is $\,dr\,d\theta\,d\phi$.

Can you suggest how to correctly setup the integral in spherical co-ordinates?

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Given that the surfaces of interest are a cylinder and a cone, I would think that a transformation to cylindrical coordinates would be more suitable. Indeed, I would shift the surfaces by $1$ unit in the $x$ direction, centering the region of integration in the $xy$-plane to the unit circle.

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