A double integral is a type of mathematical operation that is used to calculate the volume under a three-dimensional surface. It involves taking the integral of a function over a region in a two-dimensional plane.
A single integral calculates the area under a curve in a one-dimensional space, while a double integral calculates the volume under a surface in a two-dimensional space. In other words, a single integral is a special case of a double integral.
Double integrals have many practical applications, including calculating the mass of an object with varying density, finding the center of mass of an object, and determining the work done by a varying force on an object.
A double integral is evaluated by first setting up the integral based on the given function and region, and then using either the rectangular or polar coordinate system to perform the integration. This involves breaking up the region into smaller, simpler shapes and computing the integral for each shape.
Solving double integrals can be challenging due to the complexity of the calculations involved and the need to accurately set up the integral based on the given function and region. It can also be difficult to visualize the region in question, which can make it harder to choose the appropriate coordinate system for integration.