Volume of region bounded by $z=x^2+y^2$ and $z=2x$ (triple integral)

Question

Find the volume of the region bounded by $z= x^2+y^2$ and $z=2x$. The answer is $\pi/2$.

The region is a paraboloid but I'm having problem in calculating and putting the limit.

Answer 1

The integral region in the $xy$-plane is given by $x^2+y^2=2x$, a circle as seen in the form $(x-1)^2+y^2=1$. Recenter the circle with $u=x-1$ and $v=y$ to transform the region into the unit circle $u^2+v^2=1$. Then, the two surfaces become

$$z_1= 2(u+1),\>\>\>\>\>z_2=(u+1)^2+y^2$$

Integrate in cylindrical coordinates,

$$\int_{u^2+v^2\le1}(z_1-z_2)dudv =\int_{u^2+v^2\le1}(1- u^2-v^2)dudv =\int_0^{2\pi}\int_0^1 (1- r^2)rdrd\theta=\frac\pi2$$