**Problem**

A *prismatoid* is a polyhedron whose vertices all lie in one or the other of two parallel planes. The perpendicular distance between the two planes is called the *height* or *altitude* of the prismatoid. The faces that lie in the parallel planes are called the *bases* of the prismatoid. The *midsection* is the polygon formed by cutting the prismatoid by a plane parallel to the bases halfway between them.

The cross-sections of a prismatoid follow a quadratic function along its height. The general formula for computing the volume of a prismoid is

where is the height, i the area of the top base, is the area of the midsection, and is the area of the bottom base. Use integration to prove why this volumetric formula is exact.

**Solution**

Let be the height of the prismatoid. For a solid with cross-sections that follow a quadratic function along its height, and

- .

Let , , and . Then it must be shown that

- .

Working out the algebra on the right-hand side yields

- .

The algebra is simple but tedious.

Now for the left-hand side (which requires more tricks).

Use the following factorizations:

- .

- .

Since the left-hand and right-hand sides for evaluating the volume are equal,

- .