Editing Pappus's centroid theorem (section)

===Proof===

A curve given by the positive function <math> f(x) </math> is bounded by two points given by:

<math> a \geq 0 </math> and <math> b \geq a </math>

If <math> dL </math> is an infinitesimal line element tangent to the curve, the length of the curve is given by:

<math display="block"> 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 </math>

The <math> y </math> component of the centroid of this curve is:

<math display="block"> \bar{y} = \frac{1}{L} \int_a^b y \, dL = \frac{1}{L} \int_a^b y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx </math>

The area of the surface generated by rotating the curve around the x-axis is given by:

<math display="block"> A = 2 \pi \int_a^b y \, dL  = 2 \pi \int_a^b y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx </math>

Using the last two equations to eliminate the integral we have:

<math display="block"> A = 2 \pi \bar{y} L </math>