Convert to polar and evaluate

Question

I have

$$z= x^2 + y^2$$ $$z=2x$$

I set them equal to get their intersection, I get

$$2x= x^2 + y^2$$

$$0= x^2 -2x +y^2$$

by completing square I get

$$y= \pm \sqrt{1-(x-1)^2}$$

I need to put this into polar coordinates and then evaluate the integral. I was trying to shift back to the origin.

Answer 1

Your circle is centered at $(1,0)$ and radius $1$ .Put $x=r\cos \theta$ and $y=r\sin \theta$ in your equation of circle .You will get $r$ bounds from there .Your $\theta$ bounds will be from $-\dfrac{\pi}{2}$ to $\dfrac{\pi}{2}$ since your circle lies in $1$st and $4$th quadrant.No need to shift origin

For further help, you can refer this playlist. Lecture 8 and 9 contains multiple integration