Element of Area

The double integral, like the single integral, has a number of applications to geometry and physics. The basic theorem which justifies these applications is the Infinite Sum Theorem. It shows how to get an integration formula by considering an infinitely small element of area.

An element of area is a rectangle ΔD whose sides are infinitesimal and parallel to the x and y axes. Given an element of area ΔD, we let

(x, y) = lower left corner of ΔD,

Δx, Δy = dimensions of ΔD,

ΔA = Δx Δy = area of ΔD.

ΔD is illustrated in Figure 12.3.1.

Figure 12.3.1: An element of area