- #1

ohlala191785

- 18

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## 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, r

^{2}= x

^{2}+y

^{2}, tanθ = y/x

## The Attempt at a Solution

So I know that the equation for the sphere is x

^{2}+y

^{2}+z

^{2}=a

^{2}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=±√a

^{2}-x

^{2}-y

^{2}, which in cylindrical coordinates is z=±√a

^{2}-r

^{2}. Therefore the limits of integration for z are -√a

^{2}-r

^{2}and +√a

^{2}-r

^{2}.

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.