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

Triple integral in spherical coordinates

  1. Feb 29, 2012 #1
    I want to check if I'm doing this problem correctly.
    1. The problem statement, all variables and given/known data
    Region bounded by [itex]x^2+y^2=4[/itex] and bounded by the surfaces z = 0, and [itex]z=\sqrt{9-x^2-y^2}[/itex].
    Set up triple integrals which represent the volume of the solid using spherical coordinates.

    2. Relevant equations
    [itex]\int\int\int_{V}\rho^2sin\phi \; d\rho d\phi d\theta[/itex]

    3. The attempt at a solution
    The shape is a cylinder with a round top.
    It seems that I have to break this into two integrals:
    From [itex]x^2+y^2=4[/itex] and [itex]z=\sqrt{9-x^2-y^2}[/itex]:
    Since [itex]z=\rho cos\phi[/itex], [itex]\rho cos\phi = \sqrt{5} \Rightarrow \rho = \frac{\sqrt{5}}{cos\phi}[/itex]
    From [itex]z=\sqrt{9-x^2-y^2}[/itex], it is a sphere of radius 3 so ρ=3
    The sphere and cylinder meets at point (y,z)=(2, √5) and the radius makes an angle with the point:
    [itex]sin\phi = \frac{2}{3}[/itex]
    [itex]\int_{0}^{2\pi}\int_{0}^{arcsin(\frac{2}{3})}\int_{0}^{3}\rho^2sin\phi \; d\rho d\phi d\theta + \int_{0}^{2\pi}\int_{arcsin(\frac{2}{3})}^{\frac{ \pi }{2}}\int_{0}^{\frac{\sqrt{5}}{cos\phi}}\rho^2 sin \phi \; d\rho d\phi d\theta[/itex]
    Thank you.
  2. jcsd
  3. Feb 29, 2012 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Hello tintin2006. Welcome to PF !

    That all looks fine to me.
  4. Feb 29, 2012 #3
    Thanks :)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook