Results 1 to 2 of 2

Math Help - Triple Coordinates

  1. #1
    Junior Member
    Joined
    Aug 2009
    Posts
    25

    Unhappy Triple Coordinates

    Use spherical coordinates to find the volume of the region above the cone z = \sqrt {x^2 + y^2} and between the hemispheres z = \sqrt {16 - x^2 - y^2} and z = \sqrt {4 - x^2 - y^2}.

    I'm having trouble finding the bounds, would it be

     <br />
\int^{2\pi} _{0} \int^{\frac{\pi}{4}} _{0} \int^{4} _{0}<br />
  d\rho  d\theta  d\phi
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Dec 2008
    Posts
    319
    In spherical coordinates, \rho is the distance from the origin, \theta is the angle of the projection on the xy-plane with the x-axis, and \phi is the angle from the z-axis. This is relating points to the Cartesian coordinate system, of course.

    Our region is a section of a cone between two spheres. Because this cone, defined by

    z=\sqrt{x^2+y^2},

    makes an angle of 45^{\circ} or \frac{\pi}{4} radians with the z-axis, our d\phi limits will be \left(0,\frac{\pi}{4}\right). Similarly, our d\theta limits will be (0,2\pi) and our \rho limits will be (2,4).

    Now, an important thing to consider is that incrementally-defined volume regions in spherical coordinates grow larger as \rho grows larger, and also grow smaller as \phi decreases to 0 (as the volume regions shrink to wedges near the z-axis). Nothing like this happens with uniformly distributed Cartesian coordinates. To make up for it, we introduce the factor \rho^2\sin \phi when calculating triple integrals in spherical coordinates:

    \int_0^{\pi/4}\int_0^{2\pi}\int_2^4 \rho^2\sin\phi\,d\rho\,d\theta\,d\phi.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Triple integral, spheric coordinates
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 18th 2010, 10:22 PM
  2. Triple Integral in Cylindrical Coordinates
    Posted in the Calculus Forum
    Replies: 4
    Last Post: November 25th 2009, 01:37 AM
  3. Triple integral with cylindrical coordinates
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 8th 2009, 10:58 PM
  4. Triple Integral Using Spherical Coordinates
    Posted in the Calculus Forum
    Replies: 5
    Last Post: August 30th 2008, 09:50 PM
  5. Triple Integral, Spherical Coordinates
    Posted in the Calculus Forum
    Replies: 9
    Last Post: April 5th 2008, 04:44 PM

Search Tags


/mathhelpforum @mathhelpforum