Editing Spherical cap (section)

===Deriving the volume and surface area using calculus ===
[[File:Spherical cap from rotation.svg|thumb|Rotating the green area creates a spherical cap with height <math>h</math> and sphere radius <math>r</math>.]]
The volume and area formulas may be derived by examining the rotation of the function 
:<math>f(x)=\sqrt{r^2-(x-r)^2}=\sqrt{2rx-x^2}</math> 
for <math>x \in [0,h]</math>, using the formulas the [[Surface of revolution|surface of the rotation]] for the area and the [[Solid of revolution|solid of the revolution]] for the volume.
The area is
:<math>A = 2\pi\int_0^h f(x) \sqrt{1+f'(x)^2} \,dx </math>
The derivative of <math>f</math> is
:<math>f'(x) = \frac{r-x}{\sqrt{2rx-x^2}} </math>
and hence
:<math>1+f'(x)^2 = \frac{r^2}{2rx-x^2} </math>
The formula for the area is therefore
:<math>A = 2\pi\int_0^h \sqrt{2rx-x^2} \sqrt{\frac{r^2}{2rx-x^2}} \,dx
         = 2\pi \int_0^h r\,dx
         = 2\pi r \left[x\right]_0^h
         = 2 \pi r h </math>
The volume is
:<math>V = \pi \int_0^h f(x)^2 \,dx
         = \pi \int_0^h (2rx-x^2) \,dx
         = \pi \left[rx^2-\frac13x^3\right]_0^h
         = \frac{\pi h^2}{3} (3r - h)</math>