Volume lying inside the cylinder $x^2+y^2-2x=0$ and outside the parabola $x^2+y^2=2z$ while bounded by the x-y plane.

Question

Alternate/ easy to follow solution:

Change of coordinates as $(x-1)=r\cos{\theta}$ and $y=r\sin{\theta}$

Then compute the following integral: $$\iint \frac{1}{2}(x^2+y^2)dA$$ $$=\int^{2\pi}_{0}\int^{1}_{0}(r^2+2r\cos{\theta+1)}dr\cdot d\theta$$ $$=\frac{3{\pi}}{4}$$

Answer 1

Alternate/ easy to follow solution:

Change of coordinates as $(x-1)=r\cos{\theta}$ and $y=r\sin{\theta}$ as the centre of the circle is xy-plane is offset to $(1,0)$

Then compute the following integral: $$\iint \frac{1}{2}(x^2+y^2)dA$$ $$=\int^{2\pi}_{0}\int^{1}_{0}(r^2+2r\cos{\theta+1)}dr\cdot d\theta$$ $$=\frac{3{\pi}}{4}$$