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Multiple integrals for finding volume

  1. Jul 8, 2012 #1
    How do I solve this? How do I determine the range? Ill they be triple integrals?Please explain to me.

    Find the volumes in R3.

    1. Find the volume U that is bounded by the cylinder surface x^2+y^2=1 and the plane
    surfaces z=2, x+z=1.

    2. Find the volume W that is bounded by the cylindrical surfaces
    x^2 + y^2 = 1 and x^2 + z^2 = 1.

    I know these are 2 questions and they're both about finding volumes but the second one seems to have 2 cylinders :confused: ??I don't get it.
     
  2. jcsd
  3. Jul 8, 2012 #2

    tiny-tim

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    hi aruwin! :smile:

    these integration problems are basically about slicing the volume in a convenient way

    for the first one …
    … horizontal slices seem sensible:

    what is the shape of the horizontal slice at a general height z ?
    yes, two cylinders

    again, try horizontal slices, write the second condition as x2 = 1 - z2, and remember the RHS is a constant (for that slice) :wink:
     
  4. Jul 8, 2012 #3
    Horizontal slice?You mean, like a triangle?? And what do you mean by RHS is a constant?:confused:
     
  5. Jul 8, 2012 #4

    tiny-tim

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    won't one side be curved? :confused:
    in x2 = 1 - z2, z is constant for any particular slice
     
  6. Jul 8, 2012 #5
    LOL,of course one side should be a curve. My bad, I meant horizontal slice as in a slanting plane. Is that what you refer to as horizontal slice?
     
  7. Jul 8, 2012 #6

    tiny-tim

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    no, a horizontal slice means between the horizontal planes at heights z and z+dz
     
  8. Jul 8, 2012 #7
    Can you show me a diagram for the first cylinder?I am guessing it looks like a cylinder with the one slanting surface but I can't picture the XYZ coordinate inside it. I really need to see a picture.
     
  9. Jul 9, 2012 #8

    tiny-tim

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    yes, it is a cylinder with one slanting surface :smile:

    but the easiest way of integrating is to divide it into horizontal slices (so each slice is a "damaged" circle with a bit sliced off)

    you could integrate by dividing it into slanting slices, but

    i] you'd have to define a new parameter x+z, and integrate with respect to that

    ii] you'd need to know the formula for the area of an ellipse (ok, i admit that's fairly easy)

    iii] you'd still have a problem with "damaged" circles at the bottom!​
     
  10. Jul 10, 2012 #9

    haruspex

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    Before leaping into calculus, think what this shape looks like. What shape might you make with two of them?

    I think this one would be much easier using vertical slices in the x-z plane. The slices would be squares.
     
  11. Jul 10, 2012 #10

    tiny-tim

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    oooh yes … horizontal slices work, but vertical slices are easier! :smile:
     
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