Editing Pappus's centroid theorem (section)

===Proof 1===

The area bounded by the two functions:

<math display="block"> y = f(x) , \, \qquad y \geq 0 </math>

<math display="block"> y = g(x) , \, \qquad f(x) \geq g(x) </math>

and bounded by the two lines:

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

is given by:

<math display="block"> A = \int_a^b dA  = \int_a^b [f(x) - g(x)] \, dx </math>

The <math> x </math> component of the centroid of this area is given by:

<math display="block"> \bar{x} = \frac{1}{A} \, \int_a^b x \, [f(x) - g(x)] \, dx </math>

If this area is rotated about the y-axis, the volume generated can be calculated using the shell method. It is given by:

<math display="block"> V = 2 \pi \int_a^b x \, [f(x) - g(x)] \, dx </math>

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

<math display="block"> V = 2 \pi \bar{x} A </math>