Compute the volume of the shape using cylindrical coordinates. In this case the x axis is the central line, rather than the z axis, and y plays the role of r. The following double integral is taken over the region b.
Volume = 2π ∫∫y
Let a be the area of the region b, and divide through by a. Now the right side becomes 2π times the y coordinate of the centroid of b. If c represents the centroid of b, and a the area, we have the following.
v = 2πc×a
In other words, the volume of the shape is its cross sectional area times the circumference of the circle determined by the spinning centroid. If a torus has major radius q and minor radius r, its volume is 2πq×πr2.
This theorem can be run in reverse to find the centroid. If b is half the unit disk, resting on the x axis, the volume is that of the unit sphere, or 4π/3. Divide this by 2π, and by the area of the half disk to get 4 over 3π. This agrees with our earlier calculations.
We've computed the volume of our shape of revolution; what is its surface area? As you might guess, there is a similar formula. Multiply the perimeter by the circumference of the circle determined by the centroid. Thus the surface area of the torus is 2πq×2πr.
Assume the perimeter of b is piecewise differentiable. Let c(t) be a curve that traces this perimeter, and is parameterized by arc length. Thus time and distance are synonymous. The perimeter equals the travel time.
At any time t, the curve contributes 2πy(t) to the surface area. The curve may be sloping up or down, relative to the x axis, but it is parameterized by arc length, so we don't have to worry about the slope of the surface. Adding δ to t advances the curve through a distance of δ, and creates a thin band with surface area δ×2πy(t). Therefore the integral of y(t) can be used to find the surface area.
Area = 2π ∫y(t)
Divide through by the perimeter and find the centroid of b. Thus 2π times the centroid times the perimeter gives the surface area.
Run this in reverse to find the centroid of a semicircle, an arc of 180°. Divide the area of the sphere, 4π, by 2π, and then by the length of the arc, giving a centroid of 2/π.
These formulas were discovered by Pappus (biography) around 300 A.D. Thus they are sometimes called Pappus' theorems. Amazing that such beautiful theorems should predate formal calculus by 1400 years.