# Cylindrical coordinates, finding volume of solid

1. Jan 27, 2013

### ohlala191785

1. The problem statement, all variables and given/known data
Find the volume of the solid that the cylinder r = acosθ cuts out of the sphere of radius a centered at the origin.

2. Relevant equations
Cylindrical coordinates: x = rcosθ, y = rsinθ, z=z, r2 = x2+y2, tanθ = y/x

3. The attempt at a solution
So I know that the equation for the sphere is x2+y2+z2=a2 since it's centered at the origin and has a radius of a. And I'm pretty sure the integrand is 1, so the integral should look like ∫∫∫rdzdrdθ. For the limits of z, I solved for z in the sphere equation and got z=±√a2-x2-y2, which in cylindrical coordinates is z=±√a2-r2. Therefore the limits of integration for z are -√a2-r2 and +√a2-r2.

The limits of r should be from 0 to a since a is the radius of the sphere. But I have no idea what the limits of θ are. Is it from 0 to 2π because it is a sphere?

2. Jan 27, 2013

### Staff: Mentor

The limits of your cylinder are different, and depend on r. You can draw a sketch of the r,θ-plane to see this.
If cylindrical coordinates do not work, try cartesian coordinates. The integration limits are easier there.

3. Jan 28, 2013

### Coelum

You can state your problem as a double integral, where the integrand is a function of the surface at point (x,y) - or (r,θ). Can you see that function?
Beware: as stated, the cylinder is defined only for x>0 (where cosθ>0).

4. Feb 5, 2013