Page 72 - Vector Analysis

Chapter 3
Multiple Integrals

3.1 Integrable Functions

Let us start our discussion on the integrability of functions of two variables.

Definition 3.1. Let A Ď R2 be a bounded set. Define

a1 = inf ␣x P R ˇ (x, y) P A for some y P R(,
ˇ

b1 = sup ␣x P R ˇ (x, y) P A for some y P R(,
ˇ

a2 = inf ␣y P R ˇ (x, y) P A for some x P R(,
ˇ

b2 = sup ␣y P R ˇ (x, y) P A for some x P R(.
ˇ

A collection of rectangles P is called a partition of A if there exists a partition Px of [a1, b1]
and a partition Py of [a2, b2],

Px = ␣a1 = x0 ă x1 ă ¨ ¨ ¨ ă xn = b1( and Py = ␣a2 = y0 ă y1 ă ¨ ¨ ¨ ă ym = b2( ,

such that

P = ␣∆ij ˇ ∆ij = [xi, xi+1] ˆ [yj , yj+1] for i = 0, 1, ¨ ¨ ¨ , n ´ 1 and j = 0, 1, ¨ ¨ ¨ , m ´ 1( .
ˇ

The mesh size of the partition P, denoted by }P} and also called the norm of P, is defined
by

!b ˇ )

}P} = max (xi+1 ´ xi)2 + (yj+1 ´ yj )2 ˇ i = 0, 1, ¨ ¨ ¨ , n ´ 1, j = 0, 1, ¨¨ ¨ , m ´ 1 .
ˇ

The number a(xi+1 ´ xi)2 + (yj+1 ´ yj)2 is often denoted by diam(∆ij), and is called the
diameter of ∆ij.

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