Results 1 to 3 of 3

Math Help - Triple integral over the sphere centered at (1,1,1) and radius 2

  1. #1
    Member
    Joined
    Oct 2010
    Posts
    150

    Triple integral over the sphere centered at (1,1,1) and radius 2

    Problem:

    Let f(x,y,z)= xy+yz. Evaluate the triple integral of the function over the sphere centered at (1,1,1) and radius 2.

    Attempted solutions:

    I'm having trouble visualizing what this question is asking. So far I've done this:

    1. Decided I'm probably going to use spherical coordinates given I'm working with a sphere.

    2. Converted the integral to spherical coordinates.

    Questions:

    1. How do I go about describing the given sphere in spherical coordinates? I know how to describe the unit sphere, but I can't figure out how to describe spheres that have been transformed (such as this one).

    2. Am I finding the volume beneath xy+yz bounded below by the sphere?


    Any other tips would be great! Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,977
    Thanks
    1121
    First "beneath xy+ yz" makes no sense. In order to have a surface that must be equal to something. What is it equal to?

    The equation of the sphere with center (1, 1, 1) with radius 2 is (x- 1)^2+ (y- 1)^2+ (z- 1)^2= 4.

    In spherical coordinates, x= \rho cos(\theta)sin(\phi), y= \rho sin(\theta)sin(\phi), z= \rho cos(\phi). Put those into the equation of the sphere and plane.

    However, you might find it simpler to shift the entire problem so that the center of the sphere is at the origin. That is, let u= x- 1, v= y- 1, w= z- 1 so that in the uvw- coordinate system the sphere is given by u^2+ v^2+ w^2= 4 and the surface xy+ yz= f(x,y,z) becomes (u+1)(v+1)+ (v+1)(w+1)= f(u+1,v+1,w+1) (here f is the missing part of the equation of the surface).
    Now replace u, v, and w with \rho cos(\theta)sin(\phi), \rho sin(\theta)sin(\phi), and \rho cos(\phi).

    (Depending on exactly how the lower surface is defined, you might find it simpler to use cylindrical coordinates.)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2010
    Posts
    150
    "First "beneath xy+ yz" makes no sense. In order to have a surface that must be equal to something. What is it equal to?"

    In the context of the problem it would be equal to "W" right? EDIT: ah you mean equal to a constant. nvm.

    Thanks for the tips I will try them
    Last edited by divinelogos; November 15th 2010 at 08:36 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Finding radius of the sphere
    Posted in the Geometry Forum
    Replies: 9
    Last Post: January 5th 2012, 05:43 PM
  2. Replies: 1
    Last Post: March 20th 2010, 04:59 PM
  3. Radius of a sphere
    Posted in the Geometry Forum
    Replies: 4
    Last Post: July 14th 2009, 09:02 PM
  4. Find Radius of a Sphere in Vector
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 5th 2009, 01:27 PM
  5. Find Radius and Center of Sphere
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: October 12th 2008, 09:23 PM

Search Tags


/mathhelpforum @mathhelpforum