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Triple Intergral Bounds Help

  1. Jun 10, 2008 #1
    Hi guys,
    Im currently revising for my exams and I encountered a problem I hope someone will be able to help me with.

    1. The problem statement, all variables and given/known data
    Find the volume of the region of space bounded by:
    The planes x=0, y=0, z=0, z=3-2x+y and the surface y=1-x^2

    2. Relevant equations
    [tex]\int \int \int _R 1\, dV[/tex]

    3. The attempt at a solution
    First I decided to integrate with respect to the z direction as I wouldn't have to worry about splitting up the region yet.

    [tex]\int \int \int _0 ^{3-2x+y} 1\, dz\, dy\, dx[/tex]

    [tex]= \int \int 3-2x+y\, dy \, dx[/tex]

    ok. But now I have a problem due to the surface y=1-x^2 cutting our region defined by the 4 planes. Can we split the region and choose our bounds like below?

    [tex]= \int _1 ^{3/2} \int _{1-x^2} ^{2x-3} 3-2x+y\, dy\, dx \; + \int _0 ^1 \int _0 ^{2x-3} 3-2x+y\, dy \, dx[/tex]

  2. jcsd
  3. Jun 10, 2008 #2
    i have something simpler:

    say f(x,y) = 3-2x+y

    so now volume is

    int (0,1) . int (0, 1-x^2) f(x,y) dy.dx
  4. Jun 10, 2008 #3


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    Homework Helper

    If you can picture the region, the volume you want appears to be confined to the 2nd octant. For this, you'll want the limits of integration for y to be from 0 to 1-x^2. Why is any "splitting of region" necessary?
  5. Jun 10, 2008 #4
    Thanks for the reply rootX, Defennder

    Indeed you are both right.
    After scratching my head for a while I noticed I drew my diagram slightly wrong (I had the plane as z=3+2x-y lol) so my region projected onto the xy-plane was piece-wise defined.
    Last edited: Jun 10, 2008
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