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Limits of integration

  1. Mar 18, 2009 #1
    The question is this:

    Consider the tetrahedron which is bounded on three sides by the coordinate planes and on fourth side by plane x+(y/2)+(z/3)=1

    I think the region to integrate over should appear in R^2 as a right triangle, is this correct?

    Secondly i am having much trouble finding limits of integration for a double integral, can ne one help
     
  2. jcsd
  3. Mar 18, 2009 #2

    tiny-tim

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    Hi Derill03! :smile:

    For fixed x and y, what does z vary between?

    For fixed x, what does y vary between? :wink:
     
  4. Mar 18, 2009 #3
    I get a double integral integrating dydx of the function (3-3x-(3y/2)) from dy|0 to 2-2x and dx|0 to 1

    so that would leave a volume of 1
     
  5. Mar 18, 2009 #4

    tiny-tim

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    Looks good! :smile:
     
  6. Mar 18, 2009 #5
    On the next part it says to use vector methods to check the integration answer, can you point me in the right direction as to how do i use vector methods to calculate volume?

    the only volume formula i know using vectors is triple scalar product (a dot (b cross c).
     
  7. Mar 18, 2009 #6

    tiny-tim

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    That'll do! :biggrin:

    Area of triangle = 1/2 (a x b)

    Area of pyramid = 1/6 (a x b).c :wink:
     
  8. Mar 18, 2009 #7
    when i do the triple scalar product i get 6 as an answer? the integration way and vector way dont agree, any thoughts on what is wrong?
     
  9. Mar 18, 2009 #8

    tiny-tim

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    D'oh! :rolleyes:Area of pyramid = 1/6 (a x b).c
     
  10. Mar 18, 2009 #9
    I understand that if you take 1/6 of 6 you will get 1 which does agree, but its confusing to me why an area formula for a pyramid would prove a correct volume of a tetrahedron? Can you explain a little bit why this works

    Is it safe to assume that a tetrahedron and a pyramid are geometrically the same?
     
  11. Mar 18, 2009 #10

    tiny-tim

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    Pyramid is easier to write :wink:
    oh :rolleyes:

    I copied-and-pasted the other formula, and forgot to change "area" to "volume" :smile:
     
  12. Mar 18, 2009 #11
    now it all makes sense, thank you very much you were a big help
     
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