Calculating the surface area of a spherical cap using cylindrical coordinates

Question

I am trying to calculate the surface area of the spherical cap : $x^2+y^2+z^2=R^2$ and $z\ge h$. I parameterized the surface as $$x=\sqrt{R^2-z^2}\cos(\phi)$$ $$y=\sqrt{R^2-z^2}\sin(\phi)$$ $$z=z$$ I calculated the surface element as $\sqrt{(R^2-z^2)\sin^2(\phi)+(R^2-z^2)\cos^2(\phi)+z^2}$ which is equal to $R$. I am evaluating the integral $\int_{0}^{2\pi}\int_{h}^{R}Rdzd\phi$ and it is equal to $2\pi R(R-h)$. I think this equation gives me the surface area of a cylinder with radius $R$ and height $R-h$. In case of a sphere with radius $R=2$ and $z=1$, i get the surface area of the spherical cap as $4\pi$ (which is correct I think). I don't know if this is coincidental or is it the right way to calculate the surface area of a spherical cap in cylindrical coordinates. Can someone please enlighten me about the situation?

Answer 1

your calculations are not wrong. since there are too many good references and illustrations on this problem, i'm gonna put some references here.

• this video is very good illustration of this problem
• you can follow Wikipedia calculation on this problem
• This question is very related