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Finding the volume of a solid.

  1. Apr 29, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the volume of the solid in the first octant bounded by the coordinate planes and the plane 2x+y-4=0 and 8x-4z=0.


    This is a problem for a practice exam for my calculus course and I just need some help getting started.

    I have had a lot of trouble in this course trying to figure what the bounds of my integration should be so any pointers would be appreciated!

    P.S. I have also had a lot of trouble reversing the oder of integration and changing to spherical and cylindrical coordinates. Mainly because I have trouble figuring out how to change the bounds.
     
    Last edited: Apr 29, 2010
  2. jcsd
  3. Apr 29, 2010 #2

    Mark44

    Staff: Mentor

    This is not surprising, since for many problems of this kind, finding the bounds of integration is the hardest part.

    Have you drawn a picture of the solid? Drawing a picture should give you a good idea of what the region of integration looks like, and should help you get the limits of integration.
     
  4. Apr 29, 2010 #3

    lanedance

    User Avatar
    Homework Helper

    i wouldn't be using spherical or cylindrical, but would have a think about the volume - and try and draw it... whats you attempts at you bounds?
     
  5. May 2, 2010 #4
    Hey Mark,

    Thanks for the suggestion, I see what you are saying but let me ask this.

    I have a problem here where I needed to change a triple integral from cylindrical coordinates to Cartesian. I have attached an image of the problem and the solution. When I went to solve it I sketched the solid and then attempted to write the bounds of each variable. I had come to the conclusion that y should be from [tex]\sqrt{1-x^2}[/tex] to 1 which is close but not quite right. How would I come to the conclusion they made instead of what I did?
     

    Attached Files:

  6. May 2, 2010 #5
    Wow, that image would not help me learn ANYTHING!
    The limits for "r" and theta will help you determine the limits for x and y.
    Since theta only ranges from 0 to π and r ranges from 0 to 1, it is a SEMIcircle of radius one (the top half, actually). To integrate the top of the unit circle "dydx"...

    The upper limit for a vertical representative rectangle will range from y = 0 (the x-axis) to the curve y = +√(1-x^2). (...I included the "+" to emphasize that it's the top half).

    Then x ranges from -1 to 1.
     
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