Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook