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Multivariable Calculus - Integration Assignment 1#

  1. Aug 13, 2014 #1
    1. The problem statement, all variables and given/known data
    Evaluate the integral,

    [itex] \iiint_E z dzdydz [/itex]

    Where E is bounded by,

    [itex] y = 0 [/itex]
    [itex] z = 0 [/itex]
    [itex] x + y = 2 [/itex]
    [itex] y^2 + z^2 = 1 [/itex]

    in the first octant.


    2. Relevant equations

    Rearranging [itex] y^2 + z^2 = 1 [/itex] it terms of [itex] z [/itex],
    [itex] z = \sqrt{1-y^2} [/itex]


    3. The attempt at a solution

    From the given equations I determined that my bounds were,

    [itex] 1 \leq x \leq 2 [/itex]
    [itex] 0 \leq y \leq 1 [/itex]
    [itex] 0 \leq z \leq \sqrt{1-y^2} [/itex]

    I found these bounds by first looking at [itex] z = \sqrt{1-y^2}[/itex] and seeing that [itex] y [/itex] must be between 0 and 1 since we are working in the first octant, also [itex] z [/itex] must be between 0 and [itex] z = \sqrt{1-y^2}[/itex]. Then I moved on to [itex] x + y = 2 [/itex], since [itex] y [/itex] can only be between 0 and 1 the only way for the equation [itex] x + y = 2 [/itex] to be true is if [itex] x [/itex] is between 1 and 2.

    [itex] \int_1^2 \int_0^{2-x} \int_0^\sqrt{1-y^2} z dzdydz [/itex]

    After integrating I found my answer to be 1/3. Can anyone let me know if I've made a mistake anywhere or if I have done this correctly?
     
    Last edited: Aug 13, 2014
  2. jcsd
  3. Aug 13, 2014 #2
    I'm doing the same assignment. I also got 1/3

    EDIT: I'm not so sure about that answer anymore
     
    Last edited: Aug 14, 2014
  4. Aug 14, 2014 #3

    HallsofIvy

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    I get that x must be between 0 and 1. Otherwise it won't be under the cylinder [tex]y^2+ z^2= 1[/tex].
    [tex]\int_0^1\int_0^{2- x}\int_0^{\sqrt{1- y^2}} zdzdydx[/tex]
     
    Last edited: Aug 14, 2014
  5. Aug 14, 2014 #4

    Zondrina

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    Homework Helper

    Indeed, if you project the three planes and cylinder into the positive octant, you can observe that ##0 \leq x \leq 1##. You can check this out in the images I attached to help visualize. Try to see how the planes cut the cylinder, this is what lets you determine your limits most of the time.
     

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