1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    mfb

    User Avatar
    2016 Award

    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!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook