Intermediate Calculus pp 295-357 | Cite as

# Multiple Integration

## Abstract

Let *F* be a region of area *A* situated in the *xy* plane. We shall always assume that a region includes its boundary curve. Such regions are sometimes called closed regions in analogy with closed intervals on the real line that is, ones which include their endpoints. We subdivide the *xy* plane into rectangles by drawing lines parallel to the coordinate axes. These lines may or may not be equally spaced (Fig. 5-1). Starting in some convenient place (such as the upper left-hand corner of *F)*, we systematically number all the rectangles *lying entirely within F.* Suppose there are *n* such and we label them *r* _{1} *r* _{2},*…*,*r* _{n} *..* We use the symbols *A(r* _{1} *)*, *A(r* _{ 2 } *)*,*…*, *A(r* _{ n } *)* for the areas of these rectangles. The collection of *n* rectangles {r_{1}, *r* _{ 2 },*…*, *r* _{ n } *}* is called a **subdivision** A of *F.* The **norm of the subdivision** denoted by
\(\left\| \Delta \right\|\), is the length of the diagonal of the largest rectangle in the subdivision Δ.

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