Page 73 - Vector Analysis

§3.1 The Double Integrals 69

Definition 3.2. Let A Ď R2 be a bounded set, and f : A Ñ R be a bounded function. For

any partition P = ␣∆ij ˇ ∆ij = (xi, xi+1) ˆ (yj, yj+1), i = 0, ¨ ¨ ¨ , n ´ 1, j = 0, ¨ ¨ ¨ , m ´ 1(, the
ˇ

upper sum and the lower sum of f with respect to the partition P, denoted by U (f, P)

and L(f, P) respectively, are numbers defined by

U (f, P) = ÿ sup f A(x, y)A(∆ij) ,

0ďiďn´1 (x,y)P∆ij

0ďjďm´1

L(f, P) = ÿ inf f A(x, y)A(∆ij) ,
0ďiďn´1 (x,y)P∆ij
0ďjďm´1

where A(∆ij) = (xi+1 ´ xi)(yj+1 ´ yj) is the area of the rectangle ∆ij, and f A is an extension
of f , called the extension of f by zero outside A, given by

f A(x) = " f (x) x P A ,
0 x R A.

The two numbers

ż

f (x, y) dA ” inf ␣U (f, P ) ˇ P is a partition of A(
ˇ

A

and ż

f (x, y) dA ” sup ␣L(f, P) ˇ P is a partition of A(
ˇ

A

are called the upper integral and lower integral of f over A, respectively. The function

żż

f is said to be Riemann (Darboux) integrable (over A) if f (x, y)dA = f (x, y)dA,

żA A

and in this case, we express the upper and lower integral as f (x, y)dA, called the double

integral of f over A. A

Similar to the case of double integrals, we can consider the integrability of a bounded
function f defined on a bounded set A Ď Rn as follows

Definition 3.3. Let A Ď Rn be a bounded set. Define the numbers a1, a2, ¨ ¨ ¨ , an and
b1, b2, ¨ ¨ ¨ , bn by

ak = inf ␣xk P R ˇ x = (x1, ¨ ¨ ¨ , xn) P A for some x1, ¨ ¨ ¨ , xk´1, xk+1, ¨ ¨ ¨ , xn P R(,
ˇ

bk = sup ␣xk P R ˇ x = (x1, ¨ ¨ ¨ , xn) P A for some x1, ¨ ¨ ¨ , xk´1, xk+1, ¨ ¨ ¨ , xn P R(.
ˇ