Pappus's centroid theorem

In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution.

The theorems are attributed to Pappus of Alexandria and Paul Guldin. Pappus's statement of this theorem appears in print for the first time in 1659, but it was known before, by Kepler in 1615 and by Guldin in 1640.

The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to the product of the arc length s of C and the distance d traveled by the geometric centroid of C: $${\displaystyle A=sd.}$$

For example, the surface area of the torus with minor radius r and major radius R is $${\displaystyle A=(2\pi r)(2\pi R)=4\pi ^{2}Rr.}$$

A curve given by the positive function ${\displaystyle f(x)}$ is bounded by two points given by:

${\displaystyle a\geq 0}$ and ${\displaystyle b\geq a}$

If ${\displaystyle dL}$ is an infinitesimal line element tangent to the curve, the length of the curve is given by:

$${\displaystyle L=\int _{a}^{b}dL=\int _{a}^{b}{\sqrt {dx^{2}+dy^{2}}}=\int _{a}^{b}{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx}$$