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Homework Help: Finding Volume by use of Triple Integrals

  1. Apr 12, 2013 #1
    1. The problem statement, all variables and given/known data
    Find the Volume of the solid eclose by y=x[itex]^{2}[/itex]+z[itex]^{2}[/itex] and y=8-x[itex]^{2}[/itex]-z[itex]^{2}[/itex]

    3. The attempt at a solution

    Well know they're both elliptic paraboloids except one is flipped on the xz-plane and moved up 8 units. Knowing this, i equated the two equations and got 4=x[itex]^{x}[/itex]+z[itex]^{2}[/itex] which is the Domain of this volume.

    I found the limits of the intergration..... x[itex]^{2}[/itex]+z[itex]^{2}[/itex][itex]\leq[/itex]y[itex]\leq[/itex]8-x[itex]^{2}[/itex]-z[itex]^{2}[/itex]......-[itex]\sqrt{y-z^{2}}[/itex][itex]\leq[/itex]x[itex]\leq[/itex][itex]\sqrt{y-z^{2}}[/itex].....-2[itex]\leq[/itex]z[itex]\leq[/itex]2

    But my problem now is what am i actually integrating. would it be x[itex]^{2}[/itex]+z[itex]^{2}[/itex]?
    Last edited by a moderator: Apr 12, 2013
  2. jcsd
  3. Apr 12, 2013 #2


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    No. The integrand would be ##1##.$$
    Vol = \iiint_R 1\, dV$$And I would suggest you using polar coordinates in the ##xz## plane after you do the ##dy## integral.
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