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Cylindrical coordinates, finding volume of solid

  1. Jan 27, 2013 #1
    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?

    Advice would be appreciated. Thanks.
  2. jcsd
  3. Jan 27, 2013 #2


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    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.
  4. Jan 28, 2013 #3
    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).
  5. Feb 5, 2013 #4
    Thanks for the advice everyone!
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