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Volumes of Revolution with e^-x

  1. Oct 26, 2014 #1

    RJLiberator

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    1. The problem statement, all variables and given/known data

    Compute the region R in the first quadrant between y=e^(-x), x=0, and y=0. Compute using shells, the volume V of solid around the y-axis.

    2. Relevant equations
    Volume =integral of bounds 2pi*radius*height

    3. The attempt at a solution

    First I drew the graph. This graph really is just a graph of e^(-x).
    I then visually rotated it around the y-axis.
    This problem seems easy enough to set up:

    Volume = Integral between 0 (lower limit) and infinity (upper limit) of 2*pi*x*e^-(x) dx
    where x = radius
    dx = width
    e^(-x) = height

    This problem was also easy to integrate using integral substitution first, and then integration by parts one time. The final equation was

    V = 2pi[-xe^(-x)-e^(-x)] from 0 to infinity.

    After calculating the simple answer is 2pi.


    My problem: For some reason, when I use this calculator: http://www.wolframalpha.com/widgets/view.jsp?id=1cd73be1e256a7405516501e94e892ac

    I get an answer of pi/2.

    Am I doing something wrong? Or is my answer/thinking correct?

    Thank you.
     
  2. jcsd
  3. Oct 26, 2014 #2

    Dick

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    To do this by the method of shells you want to integrate dy, not dx
     
  4. Oct 26, 2014 #3

    RJLiberator

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    Oh, even if it is around the y-axis? Hm. Let me try this out again. Thank you.
     
  5. Oct 26, 2014 #4

    LCKurtz

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    No, I think Dick misspoke there. Rotating about the y axis you do want dx elements for shells. I also get ##2\pi##.
     
  6. Oct 26, 2014 #5

    Dick

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    Yes, I did misspeak and 2pi is correct. pi/2 is correct if you are rotating around the x-axis.
     
  7. Oct 26, 2014 #6

    RJLiberator

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    Ah, magnificent! So my answer is then verified. This makes me overjoyed.

    I tried it the other way and it didn't work out too well, was getting -infinity :p.

    Thank you, friends, for assisting me tonight.
     
  8. Oct 27, 2014 #7

    Mark44

    Staff: Mentor

    It's not clear from what you wrote, but you should also draw a sketch of the solid of rotation. In your first graph you should include an incremental area element that will be rotated. In your second graph, you should include a sketch of the shell or disk or whatever. If you do that, you'll have a better chance of getting the integrand right, which in this case it seems that you did.
     
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