3.4. Multiple integrals and their applications 3.4.1. The concept of double integral

Let the function f ( x, y ) be defined inside some domain D and on its boundary. Divide the domain D into n partial domains D ₁ , D ₂ , ..., D _n . We denote their areas by σ ₁ , σ ₂ , ..., σ _n . The largest chord (the segment connecting the two points of the boundary of the region) of each of the regions is called its diameter. Let h denote the diameter, the largest of all n diameters. In each partial region, we take the point ( P ₁ ( x ₁ ; y ₁ ) in the region D ₁ , P ₂ ( x ₂ ; y ₂ ) in the region D ₂ , etc.). We make the integral sum:

Let n tend to infinity so that h tends to zero.

The final limit of the sequence S _n (if it exists) as h → 0 , which depends neither on the method of dividing the domain D , nor on the choice of the points P ₁ , P ₂ , ..., P _n , is called the double integral of the function f ( x, y ) and is denoted by .

The function f ( x, y ) is called an integrable function on the domain D. The region D is called the region of integration .

A continuous function on a closed domain is integrable on this domain.