Finding surface area of cone in spherical coordinates

  • Thread starter ninevolt
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  • #1
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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]
Where:
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?
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,832
251
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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