# Measuring Volume Under a Surface Using a Double Integral

A double integral allows you to measure the volume under a surface as bounded by a rectangle. Definite integrals provide a reliable way to measure the signed area between a function and the *x*-axis* *as bounded by any two values of *x**.* Similarly, a *double* *integral* allows you to measure the signed volume between a function *z* = *f*(*x**,* *y*) and the *xy*-plane as bounded by any two values of *x* and any two values of *y**.*

Here’s an example of a double integral:

Although it may look complicated, a double integral is really an integral inside another integral. To help you see this, the inner integral in the previous example is bracketed off here:

When you focus on the integral inside the brackets, you can see that the limits of integration for 0 and 1 correspond with the *dx* — that is, *x* = 0 and *x* = 1. Similarly, the limits of integration 0 and 2 correspond with the *dy* — that is, *y* = 0 and *y *= 2.

This figure shows you what this volume looks like. The double integral measures the volume between *f*(*x**,* *y*) and the *xy*-plane as bounded by a rectangle. In this case, the rectangle is described by the four lines *x* = 0, *x* = 1, *y* = 0, and *y* = 2.