Multiple Integrals

When you work with functions of more than one variable, integration becomes a powerful tool for measuring quantities like area, volume, and mass over complex regions. In single-variable calculus, you are familiar with the definite integral as the area under a curve. With multivariate functions, you extend this idea to double and triple integrals.

A double integral is used to compute the volume under a surface defined by a function $f(x, y)$ over a region $R$ in the $xy$-plane. You write this as:

where $dA$ represents an infinitesimal area element, and $R$ defines the limits of integration. The region $R$ could be a rectangle, a circle, or any more general shape.

To evaluate a double integral, you often set it up as an iterated integral, integrating with respect to one variable first, then the other. For a rectangular region where $a ≤ x ≤ b$ and $c ≤ y ≤ d$, the double integral becomes:

For more general regions, you must carefully describe the bounds for each variable, which may depend on the other variable. This is called setting up the region of integration.

A triple integral extends this concept to functions of three variables, $f(x, y, z)$, and allows you to compute quantities like mass or total charge within a three-dimensional region. The triple integral is written as:

where $E$ is a region in three-dimensional space and $dV$ is an infinitesimal volume element.

Both double and triple integrals are essential for solving real-world problems in physics, engineering, and probability, such as finding the mass of a solid with variable density or the total probability over a two-dimensional region.