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Integral Volume of Sphere

  1. Jan 14, 2010 #1
    1. The problem statement, all variables and given/known data
    For a sphere of radius r find the volume of the cap of height h.


    2. Relevant equations



    3. The attempt at a solution
    I can get it down to [tex]V \ = \ \pi \int_{r-h}^r (\pi r^2 - \pi y^2)dy \ = \ \pi {\left[(r^2y-\dfrac{1}{3} y^3) \right] }_{r-h}^r[/tex]

    I expanded this to [tex]V=\pi (-\dfrac{2}{3} r^3+2r^2h-rh^2-\dfrac{1}{3}h^3)[/tex]

    but the book has [tex]V=\pi h^2(r-\dfrac{1}{3} h)[/tex]

    What am I missing/doing wrong?
     
    Last edited: Jan 14, 2010
  2. jcsd
  3. Jan 14, 2010 #2

    Dick

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    Science Advisor
    Homework Helper

    The calculus part is ok, except that you mistyped your integrand. The problem is in the algebra leading to your expression for V. For example, I can tell by looking at your integral expression that there shouldn't be any r^3 term in the result.
     
  4. Jan 14, 2010 #3
    do the expansion again?
     
  5. Jan 14, 2010 #4
    Yeah I got it. Forgot to carry the subtraction in the [tex] (r-h)^3 [/tex] expansion twice in a row and just assumed I was missing something instead of checking my algebra again. I should have seen that the [tex] r^3 [/tex] expressions would cancel out. Sorry.
     
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