Limits of integration question (double integral)

[iScience]

so in the image in the link below, i don't understand a couple of things:

1.) the center of the cylinder is off to the side and not at the center. where/how in the problem are we taking this into account? because it should definitely affect the volume under the parabaloid right?

2.) most of the other times I've worked with circles/cylinders/spheres in polar coordinates i'ved always used 0≤θ≤2\pi as my limits of integration for theta. what's with the \frac{-\pi}{2}≤θ≤\frac{\pi}{2} limits? ie why am i only integrating half of the circle?

3.) r=2cosθ??... where did this come from?..http://i.imgur.com/THQz9Qc.jpg

(i don't know how to make the image smaller on here using the so i just posted the site)

 

[CAF123]

so in the image in the link below, i don't understand a couple of things:

1.) the center of the cylinder is off to the side and not at the center. where/how in the problem are we taking this into account? because it should definitely affect the volume under the parabaloid right?

The ranges of r and θ are affected by this shift, as you have noticed below.

2.) most of the other times I've worked with circles/cylinders/spheres in polar coordinates i'ved always used 0≤θ≤2\pi as my limits of integration for theta. what's with the \frac{-\pi}{2}≤θ≤\frac{\pi}{2} limits? ie why am i only integrating half of the circle?

It does not represent half of the circle, but rather all of it. How is θ defined?

3.) r=2cosθ??... where did this come from?..

This comes about from the projection of the intersection of the cylinder and the paraboloid onto the xy plane.

I think I had a question like this in my exam last year.