Usually, one's second year of calculus deals with applications of integration. Among them are methods for finding the volume of a solid of revolution and the surface area of a surface of revolution. Pappus of Alexandria, however, had no knowledge of calculus, and yet had ways of solving these same types of problems without falling back on integration.

Pappus' theorem for surfaces of revolution:

The surface area of a surface of revolution is equal to the the circumference of the path taken by the center of mass of the figure as it revolves, multiplied by the outer perimeter of the generating figure.

Pappus' theorem for solids of revolution:

The volume of a solid of revolution is equal to the circumference of the path taken by the center of mass of the figure as it revolves, multiplied by the area of the generating figure.

For simple things like cones and other solids with easy to calculate centroids, Pappus' theorems make integration all but unecessary. If you're only concerned with an approximation, you can always use an integration by scissors technique (or perhaps integration by rope for perimeter). Otherwise, you're left with calculus for finding the center of mass — which may not seem like much of an improvement.

This technique is probably how the ancients found the surface area and volume formulas for the sphere, cone, cylinder, torus, etc. It was lost during the dark ages, only to be rediscovered by Paul Guldin. Whether or not Guldin discovered it independently or found it from the Arabian texts from which virtually all our knowledge of Greek mathematics comes from is up to debate.

How is it used?

The easiest example of using Pappus' theorem is deriving the surface area and volume of a torus. A torus is really a circle of radius r that is rotated about a larger radius R. From that node, we find the following two facts:

1. S = 4π²Rr
2. V = 2π²Rr²

The Surface Area of a Torus

The centroid of a circle is its center, conveniently enough. This means that it travels about a circumference of 2πR. The circumference of the circle is 2πr. By Pappus' Theorem, then, S = 2πR⋅2πr = 4π²Rr, as expected.

The Volume of a Torus

Again, the path traveled by the centroid of the generating circle is 2πR, but this time the area of the circle is the familiar formula πr². By Pappus' Theorem again, V = 2πR⋅πr² = 2π²Rr².

Of course, with more complicated shapes, finding the centroid of the figure usually means resorting to calculus anyways.