Iterated Integral

In this section we shall learn how to evaluate double integrals. A double integral can be evaluated by two single integrations. The Iterated Integral Theorem gives the key formula.

The iterated integral

is an integral of an integral of f(x, y). It is evaluated in two stages. First evaluate the inside integral

by ordinary definite integration, treating x as a constant. This gives us a function of x alone. Second, evaluate the outside integral

by a second definite integration.

We shall usually drop the brackets around the inside integral and write the iterated integral as