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Finding surface area of cone in spherical coordinates

  1. Jan 26, 2012 #1
    Hello everyone,

    I recently tried to find the surface area of a hollow cone (there is no base, like an ice cream cone) using spherical coordinates. With cylindrical coordinates I was able to do this easily using the following integral:

    [itex]\int \int \frac{R}{h}z \sqrt{\frac{R^{2}}{h^{2}} + 1} dz d\theta[/itex]
    R = radius of the base
    h = height of the cone
    (R/h)z = radius of cone at specific z

    [itex]\sqrt{\frac{R^{2}}{h^{2}} + 1}[/itex] - the ds element across the slanted side of the cone

    and I will obtain the correct answer for the surface area of a cone:
    [itex]\pi R \sqrt{h^{2} + R^{2}}[/itex]

    but when I try to do the same integral in spherical coordinates I obtain different results
    I use the following integral:
    [itex]\int \int \rho^{2} sin(\theta) d\rho d\phi [/itex]

    What am I doing wrong?
  2. jcsd
  3. Jan 27, 2012 #2


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    Science Advisor
    Homework Helper

    hello ninevolt! :smile:

    i think you're confusing θ with the (fixed) semi-angle of the cone :wink:

    (btw, you might also like to try doing it without integration, by slicing the cone and flattening it!)
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