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Double Integration to Find Volume

  1. Nov 10, 2007 #1

    Find the volume of the region inside the surface z=x^2+y^2 and between z=0 and z=10

    2. Relevant equations


    3. The attempt at a solution

    I know that I have to use some sort of double integration to find this volume, but I'm not really sure where to begin with the problem
  2. jcsd
  3. Nov 10, 2007 #2
    [tex]\iiint_V dV[/tex] the upper curve is [tex]z=10[/tex] the lower curve is [tex]z=0[/tex] the area of integration in [tex]\mathbb{R}^2[/tex] is a circle of radius 10. Now use cyclindrical change of variable.
  4. Nov 10, 2007 #3
    hmm...I may try that if I can't find an alternative, but is there any way to do this problem with a double integration and not a triple integration? because the section i'm working with is strictly double integration
  5. Nov 11, 2007 #4


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    Staff Emeritus
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    That's a paraboloid. Since you have "flat" bottom and top I recommend you imagine the solid consisting of thin horizontal pieces. It should be obvious that the piece at height z is a disk satisfying [itex]x^2+ y^2= z[/itex]. What is the area of that disk? If you think of the disk as having thickness "dz", what is its volume? Now "add" the volumes of all those disks.

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