Results 1 to 2 of 2

Thread: Mass of a solid

  1. #1
    Junior Member
    Nov 2008

    Mass of a solid

    Right now I'm learning about the Change of Variables formula and applications of integration, and I'm having trouble with this question:

    Find the mass of the solid bounded by the cylinder x^2+y^2=2x and the cone z^2=x^2+y^2 if the density is delta=sqrt(x^2+y^2).

    I have the idea that I should switch to cylindrical coordinates. So I would let x=rcostheta, y=rsintheta, and z=z. I get that delta=r, but I am having trouble figuring out what my limits of integration will be. Any help would be appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Apr 2005
    Yes, I agree that, because of the circular symmetry, cylindrical coordinates should simplify the integration. I assume that by "the solid bounded by" the cylinder and cone, they mean the region between the two "nappes" of the cone, inside the cylinder.

    The easy part- \theta will range from 0 to 2\pi. Slightly harder: Since the cone, in cylindrical coordinates, is z^2= r^2 and the cylinder by r= 2cos(\theta) ( x^2+ y^2= r^2 and 2x= 2rcos(\theta): r^2= 2rcos(\theta) and we can cancel an "r"), the cone and cylinder intersect when z^2= 4cos^2(\theta). For each [itex]\theta[/itex], z ranges from -2 cos(\theta) to 2 cos(\theta). Finally, for each z and \theta, r ranges from the cone out to the cylinder. The cone is r= |z| (r is positive, of course) and the cylinder r= 2cos(\theta) as before.

    Your density function is \delta= \sqrt{x^2+ y^2}= r and the differential of volume is r drd\theta dz.

    Putting all of that together, your integral is
    \int_{\theta= 0}^{2\pi}\int_{z= -2cos(\theta)}^{2cos(\theta)}\int_{r= |z|}^{2cos(\theta)} r (r drdzd\theta)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Center of mass of a solid
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Apr 4th 2011, 12:08 PM
  2. find the mass of the solid
    Posted in the Calculus Forum
    Replies: 6
    Last Post: Jan 1st 2011, 01:14 PM
  3. mass of solid, chemistry calculation.
    Posted in the Math Topics Forum
    Replies: 7
    Last Post: Nov 30th 2010, 12:29 PM
  4. Replies: 1
    Last Post: Nov 11th 2009, 05:39 PM
  5. center of mass of a solid
    Posted in the Calculus Forum
    Replies: 12
    Last Post: Dec 9th 2008, 07:39 AM

Search Tags

/mathhelpforum @mathhelpforum