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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?