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Multivariable Calculus Triple Integration Problem

  1. Apr 30, 2014 #1
    1. The problem statement, all variables and given/known data
    Express the iterated integral ∫[0,1]∫[0,1-y^2]∫[0,y] f(x,y,z)dzdxdy
    a. as a triple integral (i.e., describe the region of integration);
    b. as an iterated integral in the order z, y, x;
    c. as an iterated integral in the order y, z, x:




    3. The attempt at a solution
    so would writing (0<=z<=y) (0<=x<=1-y^2) (0<=y<=1) be sufficient for part a.

    for part b. i got ∫[0,1] ∫[0,√1-x] ∫[0,y] which i think is correct.

    for part c. however i had difficulty in changing the order because when i graph the dz and dy bounds to switch them, dy is in terms of x and I get stuck.
     
  2. jcsd
  3. Apr 30, 2014 #2

    Simon Bridge

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    (a) A triple integral usually has a triple-integration sign in it.
    You are also asked to describe the region of integration - what shape it is? Can you put it in words?
    It can help to sketch it out. Though your notes may have a standard approach to how you are supposed to describe the regions.

    Understanding this part will help you with the rest.
     
  4. Apr 30, 2014 #3
    would this be an accurate graphing?
     

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  5. Apr 30, 2014 #4

    Simon Bridge

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    Don't forget to label important parts of the sketch ... like where are the bits that are equal to one?
    Off context: you appear to have sketched the parabolic surface x=1-y^2 inside 0<z<1 ... is that correct?

    Will that give you the integral that you started with?
     
  6. Apr 30, 2014 #5
    ah i see I graphed the z component incorrectly. I'm having trouble visualizing/drawing the 0<z<y part of the graph
     
  7. May 1, 2014 #6

    HallsofIvy

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    [itex]0\le z \le y[/itex] is a plane.
     
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