Cylindrical coordinates, finding volume of solid

[ohlala191785]

Homework Statement

Find the volume of the solid that the cylinder r = acosθ cuts out of the sphere of radius a centered at the origin.

Homework Equations

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

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?

Advice would be appreciated. Thanks.

 

[mfb]

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.

 

[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).

 

[ohlala191785]

Thanks for the advice everyone!