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Homework Help: Setting up a double integral to find the volume

  1. Jun 27, 2010 #1
    1. The problem statement, all variables and given/known data

    Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equation

    x2+y2+z2=r2



    2. Relevant equations
    Not much equations, just setting the integral up, however I have no idea.


    3. The attempt at a solution

    I know how to approach these problems if there were only 2 variables, but I'm kind of stuck since there are three variables that I have to deal with.
     
  2. jcsd
  3. Jun 27, 2010 #2

    LCKurtz

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    Then solve the equation for z, giving one or more functions of two variables.

    Is there any obvious symmetry you can use?
     
  4. Jun 27, 2010 #3
    how can you solve the equation for z, even then there will be r^2. Help me out with the beginning here please.
     
  5. Jun 27, 2010 #4

    LCKurtz

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    r is just a constant. Do you recognize what the graph of this is?
     
  6. Jun 27, 2010 #5
    do we approach this problem by first saying z=o, then x=o and y=o, integrating all three equations. However the question says to set up a double integration, not a triple integration?
     
  7. Jun 27, 2010 #6

    LCKurtz

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    No, you don't do that. So I will ask you again:

    1. Do you recognize what this surface is?
    2. Can you use any symmetries to your advantage?

    Then solve it for z to get started. Get z in terms of x and y if you are required to do a double integral.
     
  8. Jun 27, 2010 #7
    well it is a sphere, we can say that the center of the sphere passes through (0,0,0) so if we calculate the top half, we can multiply by 2, to get the answer.

    z=sqrt(r^2-x^2-y^2)

    the limits being -r and r for the first integral, and sqrt(r^2-y^2) and -sqrt(r^2-y^2)

    is this correct so far?
     
  9. Jun 28, 2010 #8

    LCKurtz

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    You have the right idea. You might want to change your dxdy integral to polar coordinates to make it easier. If you do that, you might first change the r in the equation of the sphere to a so you don't confuse it with the r in polar coordinates. Good luck. I'm off to bed.
     
  10. Jun 28, 2010 #9
    x2+y2+z2=r2 is a sphere..
    try using spherical coordinates.
    set up a triple integral, and do one integration to get to the double integral :)
     
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