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Volume of F over R

  1. Nov 3, 2011 #1
    1. The problem statement, all variables and given/known data
    Find the volume of the solid that is bounded on the front and back by the planes x = 1 and x = 2, on the sides by the cylinders y = +/-(1/x) and below by the planes z = x + 1 and z = 0


    3. The attempt at a solution
    I can't picture this thing at all, my sketch is so convoluted I am having trouble finding my bounds. If there's an easier way to get around this, I would love to see how. I do know that I have to use a double integral to compute V.
     
  2. jcsd
  3. Nov 3, 2011 #2
    The best way is to just flat-out draw them in Mathematica. Otherwise, just need to practice drawing them. Make like only the picture is the homework and you are required to draw a nice one. Spend time doing just that and don't worry about the integral just yet. Then, easier to visualize this as a triple integral:

    [tex]\int\int\int dzdydx[/tex]

    which reduces to a double since the integrand is just f(x,y,z)=1. So to integrate over z first, we integrate from some lower surface z=g(x,y), to some upper surface z=h(x,y). You said z goes from z=g(x,y)=0 to z=h(x,y)=x-1 so wouldn't we be integrating between those two surfaces? How then would we have to write the triple (or double) integral for that?

    Now, for the double integral we next integrate between two curves in x-y space, from y=f1(x) to yj=f2(x). Well, again, you said y goes from -1/x to 1/x. Ok. Finally, for the outer integral, we integrate between two points on the x-axis and you gave that as well.
     
    Last edited: Nov 3, 2011
  4. Nov 4, 2011 #3
    okay, yeah it's been answered already, it's hard to picture a lot of these and obviously i'm not going to be able to use mathematica on my final. thanks!
     
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