Iterated Integrals and Fubini's Theorem

A double integral finds the total accumulation of a function over a region. It is evaluated using an iterated integral, which expresses the double integral as two single integrals.

For a function over a rectangle in the xy-plane, holding x constant means slicing the region perpendicular to the x-axis and integrating with respect to y. This result is now a function of x, and integrating it with respect to x gives the total accumulation. This process also works in reverse: first along x, then y.

Fubini’s Theorem governs this process. It states that for a continuous function on a rectangle, both orders of integration give the same result.

This concept applies to real-world problems, such as finding the mass of a thin sheet with variable density.

In this case, integrating the density with respect to y gives the mass of a thin strip. Integrating these strip masses with respect to x gives the total mass.

Similarly, integrating with respect to x first and then y gives the same value. Fubini’s Theorem ensures that either order gives the same result, so the order of integration that makes the calculation easier can be chosen.