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Write a triple integral in spherical coordinates

  1. Apr 24, 2014 #1
    1. The problem statement, all variables and given/known data

    Write a triple integral in spherical coordinates that represents the volume of the part of the sphere
    X^2+Y^2+Z^2=16 that lies in the first octant(where x,y, and z are coordinates are all greater than or equal to zero)

    2. Relevant equations

    So i know this is in rectangular form (x,y,z) trying to get it into (p,Θ,ø)

    3. The attempt at a solution


    To get Θ the formula is
    arccos z/(square root of x^2+y^2+z^2)

    when i solve for z I get z=√(16-r^2)

    These are the limits i know
    The limits for Dz are from z=0 to z=√(16-r^2)
    The limits for Dr are from r=0 to r= pi/2

    I do not know how to find the limits for DΘ (theta)

    since the arccos √(16-r^2)/(square root of x^2+y^2+z^2)
    should give me my theta. Θ
    but i have no real values for x y and z. so i dont know how to approach this.
    thank you

    ∫ ∫ ∫
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Apr 24, 2014 #2


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    You don't seem to understand the problem. You should figure out how a differential element of volume dV is represented in spherical coordinates. In cartesian coordinates, dV = dx dy dz. You appear to have used cylindrical coordinates, r, z, and Θ.

  4. Apr 24, 2014 #3


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    Indeed, and in applying that transformation properly, you will wind up needing to solve:

    ##r sin(\theta) cos(\phi) ≥ 0##
    ##r sin(\theta) sin(\phi) ≥ 0##
    ##r cos(\theta) ≥ 0##

    Don't forget the Jacobian.
  5. Apr 24, 2014 #4


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    @JtechGuy21: Don't set that problem up in rectangular coordinates and try to change it to spherical coordinates. You don't need to change variables and mess with Jacobians. Just remember$$
    \text{Volume} =\iiint_V 1~dV$$All you need is the ##dV## formula in spherical coordinates and then to put the ##\rho,\phi,\theta## limits on. They are very easy and can be easily seen geometrically.
    Last edited: Apr 24, 2014
  6. Apr 25, 2014 #5
    aaaaaah okay.
    yes i was turning them into cylindrical for some reason...

    Thanks for pointing that out.


    1Dv=p^2sin dp dø dΘ

    p^2 Sinø dp dø dΘ

    So that means my limits for;
    p is 0 to 4
    Θ is 0 to pi/2 (because its quadrant one)
    and i have no idea how to find my ø limits?
    Since i dont have a real value for z. I cant take the arc cos of z to get my ø?
  7. Apr 25, 2014 #6


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    It would be good for you to quote who you are replying to.

    Look at the picture of the sphere in the first octant. Draw a radius and label ##\rho##, ##\phi## and ##\theta##. It is obvious from the picture what their ranges are. You have ##\rho## and ##\theta## correct so far.
  8. Apr 25, 2014 #7
    i dont think i need to use a jacobian here. im in the section that involves using triple integrals in cylindrical and spherical coordinates.(not saying its not possible )

    The section afterwards is change of variables:jacobians.
    Which I've done before, but since we are finding volume in this case, i dont know how jacobians would help since jacobians deal with area in the uv plane in comparison to the xy plane
  9. Apr 25, 2014 #8
    This might be a dumb question, but how do i draw this picture so i could visually see whats happening.
    Do i have to put it into z= format?
    and use wolfgram alpha
  10. Apr 25, 2014 #9


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    Just draw it by hand. Draw a section of a sphere that is in the first octant. Surely your book has such pictures. Probably right where it introduces spherical coordinates.

    Or look here: http://mathinsight.org/spherical_coordinates

    It isn't hard to see the limits for the variables in the first octant. Be aware that physics and math books have the ##\phi## and ##\theta## switched. That link uses the math convention.
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