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Homework Help: Integral between a cone and a cylinder

  1. Jan 23, 2012 #1
    Find the mass of the solid bounded by the cylinder (x-1)2 + y2=1 and the cone z=(x2+y2) 1/2 if the density is (x2+y2) 1/2

    I know that I have to substitute for cylindrical co-ordinates x=rcos(theta), y=rsin(theta), and z=z, and then use the change of variables formula to get the mass by integrating the density over the region. I'm just having a really hard time expressing the bounds of the region using cylindrical co-ordinates

    for the cone, bounds are
    Last edited: Jan 23, 2012
  2. jcsd
  3. Jan 23, 2012 #2


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    Hello embemilyy. Welcome to PF !

    Certainly there's more to this problem.

    What is the density?

    Please state the complete problem word for word so we may help you.
  4. Jan 23, 2012 #3
    Thanks for responding!
    Sorry about that, I just posted the density.
    density= (x2+y2)1/2.
    It is bounded between the cone, the cylinder, and the plane z=0.
    So I'm thinking the z bounds must be z=0 to z=r. I'm having trouble coming up with the bounds for r though, because of the shift in the cylinder.
  5. Jan 23, 2012 #4
    There's something not quite right here. The cylinder has a vertical axis and is the projection of the unit circle centered on (1,0) in the XY plane. The cone is a right circular cone with vertex at the origin, opening out linearly in accordance with z=r where r = sqrt(x2+y2) is the radial coordinate when using cylindrical coordinates. It seems to me that the intersection of these figures has an infinite volume. Since the density does not depend on z, it would appear that the mass is infinite, too....

    Is the intention to consider the intersection of the cylinder and the complement (outside) of the cone? That at least gives a finite answer....
  6. Jan 23, 2012 #5
    I was thinking of it as the region outside of the cone, above the xy-plane
  7. Jan 24, 2012 #6
    Any ideas?
  8. Jan 24, 2012 #7


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    write the equation of the cylinder in cylindrical coords.
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