I'm trying to understand how to find the angle for the integration in polar coordinate form for a solid. Here's an example of what I'm trying to solve: Find the volume of the solid bounded by the graphs of the given equations:

r= cos(theta), z= 2+ x^2 + y^2, z = 0.

I realize that the radius can go from 0 to cos(theta), and the function to the integrate is (2+r^2) r dr d(theta)

However, I can't seem to understand how to find the angle of integration. in the solutions, it uses symmetry and becomes 2∫from 0 to pi/2 ∫ from 0 to cos(theta) of [(2+r^2) r dr d(theta)].

The only thing I can't understand is how the angle was found, and why the integral was then multiplied by two.

Thank you very much.


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