Final answer:
To calculate a double integral, define the limits of integration for the region D and the scalar function f(x,y), then perform the integration over the area. Double integrals aggregate infinitesimal elements over a two-dimensional space. In physics, such an integral can calculate quantities like electric flux.
Step-by-step explanation:
To calculate the multiple integral ∫D f(x,y) over a region D in a three-dimensional space, we must first specify the limits of integration corresponding to the region D, and the scalar function f(x,y) to be integrated. In a three-dimensional space (xy-plane), the 'double integral' typically refers to integrating over an area, whereas for integrating over a volume, a 'triple integral' would be necessary. The scalar function f(x,y) should depend solely on x and y for us to properly evaluate a double integral. If the scalar function also depends on a third variable, say z, then we would be venturing into a triple integral.
Double integrals involve finding the sum of infinitesimal elements over a two-dimensional region. For example, in physics, calculating the electric flux through a plane might involve using a double integral to account for the plane's area, considering the electric field strength and the orientation of the plane relative to the field direction.
To evaluate such an integral, one might need to convert the multiple integral into one or more single-variable integrals. The choice of variable(s) depends on the given scenario and could be made to simplify the integration process. The example involving electric flux demonstrates the reduction of a surface integral to a double integral.