Surface Area of a Sphere
For the surface area of a sphere, examine the "globe" in the diagram. Essentially all the circles of latitude are added up with a "thickness" dS. Notice how the globe is composed of "panels." The sizes of the panels can be calculated by restricting the range of the integral equation and then cut it up as if it were a pizza.
The following illustration of the cross section through a great circle on the globe; they are the circles of "longitude." The pointer marks out dS which we spin around the globe at angle theta with a radius of x. So all the circles between negative pi divided by two to pi divided by two are added.
The Surface Area of a Spherical Cap or Dome
The surface area of a spherical cap or dome can be calculated using a similar technique as above by restricting the range of the equation. In the following illustration, we want to calculate the area of the green portion of the sphere based upon values of its height h and its radius r.
Similar to the above diagram through a great circle of "longitude," the "pointer" of ds, as integrated from negative pi divided by two to pi divided by two, is spun around the yaxis with a radius of p for each angle. We need not worry about p (x above) anymore in this derivation as we're reusing the equation above that accounts for it.
So, the question becomes at what angle of theta should we use as the lower value in the above equation for a given height (h) and radius (r)? Well, consider the following:
