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Trouble visualizing what is going on. Volume of object

  1. Sep 3, 2013 #1
    1. The problem statement, all variables and given/known data
    Hello,
    Here is a link to a pdf .http://www.mrskeller.net/documents/hwsols14.pdf
    I'm having issues with number 56. The solution makes no sense to me.



    2. Relevant equations



    3. The attempt at a solution

    So, I did graph the equation y = sqrt(r^2 - x^2) and I understand that height of the circle. I understand that you need to integrate from -r to r. But I do not understand the integral by any means.

    They have ##\int 4(r^2-x^2)) dx ## from -r to r
    I thought that maybe there were doing something of the form ∏∫y^2 because that is what this chapter uses. But I guess not. They say that the length of a side is 2y = 2sqrt(r^2 - x^2). And the only way that I can see how to get from that equation to the integral they suggest is ly squaring 2y = 2sqrt(r^2 - x^2). So I don't understand that even because if you squared it and used the integral of the form ∏∫y^2 dx then your 4 dissapears. I just don't understand what is going on with this problem. Thanks
     
  2. jcsd
  3. Sep 3, 2013 #2

    eumyang

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    Homework Helper

    The volume of a solid with a known cross section is basically
    [tex]{\int_a}^b A(x) dx[/tex],
    where A(x) represents the area of the cross section.

    I believe the attached graphic illustrates the problem. Imagine a square lying on its side, the side being the thin rectangle in the diagram. Now imagine a bunch of thin vertical rectangles going from left to right (from -r to r). The distance from the x-axis to the top of this particular rectangle is y. Then you know the length of this thin rectangle (= the length of the square cross section), which is 2y. The area of the square cross section is
    [tex]A = s^2[/tex]
    (s is the length of the side),
    so plug in 2y for s. Now, what does y equal in terms of x and r? Plug that in for y, and the resulting expression serves as the integrand for this problem.
     

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  4. Sep 3, 2013 #3
    OMG! Thank you that makes a lot more sense. A lot more!
     
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