What is a Double Integral?

A double integral is an extension of the concept of integration in calculus to functions of two variables. Double integrals are used to compute the volume under a surface or to find the total accumulated quantity over a two-dimensional region.

Double integrals are calculated by iteratively applying the single-variable integration process. This is typically done by first integrating with respect to one variable while treating the other as a constant, and then integrating the resulting expression with respect to the second variable. The order of integration can be interchanged, depending on the problem and the region’s geometry.

In some cases, it is more convenient to use polar coordinates instead of Cartesian coordinates for calculating double integrals. This involves transforming the integrand and the region of integration into polar coordinates and appropriately adjusting the integration limits.

Double integrals have numerous applications in physics, engineering, and other disciplines, such as calculating the mass of an object with variable density, finding the electric charge in a region, or determining the average value of a function over a two-dimensional area.