Page 78 - Vector Analysis

74 CHAPTER 3. Multiple Integrals

all sets of sample points tξ1, ¨ ¨ ¨ , ξN u for P, we must have

ˇ N f A(ξk)ν(∆k) ˇ ε
ˇ Iˇ 4
ˇ ÿ
ˇ
´ ă .

k=1

Let P = t∆1, ¨ ¨ ¨ , ∆N ( be a partition of A with }P} ă δ. Choose two sample sets
tξ1, ¨ ¨ ¨ , ξN u and tη1, ¨ ¨ ¨ , ηN u for P such that

(a) sup f A(x) ´ ε ă f A(ξk) ď sup f A(x);
4ν(R)
xP∆k xP∆k

(b) inf f A(x) + ε ą f A(ηk) ě inf f A(x).
4ν(R) xP∆k
xP∆k

Then

U (f, P) = N sup f A(x)ν(∆k) ă N [ A (ξk ) + ε ] (∆k)
f (R) ν
ÿ xP∆k ÿ 4ν

k=1 k=1

N f A(ξk)ν(∆k) ε N I ε ε I ε
4ν(R) 4 4 2
ÿ ÿ

= + ν(∆k) ă + + = +

k=1 k=1

and

L(f, P) = N inf f A(x)ν(∆k) ą N [ A (ηk ) ´ ε ] )
f (R) ν(∆k
ÿ xP∆k ÿ 4ν

k=1 k=1

N f A(ηk)ν(∆k) ε N I ε ε I ε
4ν(R) 4 4 2
ÿ ÿ

= ´ ν(∆k) ą ´ ´ = ´ .

k=1 k=1

As a consequence, I ´ ε ă L(f, P) ď U (f, P) ă I + ε; thus U (f, P) ´ L(f, P) ă ε.
2
2

ż

“ñ” Let I = f (x)dx, and ε ą 0 be given. Since f is Riemann integrable over A, there

A ε.

exists a partition P1 of A such that U (f, P1) ´ L(f, P1) ă 2 Suppose that P1(i) =

␣y0(i), y1(i), ¨ ¨ ¨ , ym(i)i( for 1 ď i ď n. With M denoting the number m1 + m2 + ¨ ¨ ¨ + mn,

we define

δ = ( ε ) .
n) 1
4rn´1(M + sup f A (R) ´ inf f A(R) +

Then δ ą 0. Our goal is to show that if P is a partition of A with }P} ă δ and

tξ1, ¨ ¨ ¨ , ξN u is a set of sample points for P, then (3.2) holds.

Assume that P = t∆1, ∆2, ¨ ¨ ¨ , ∆N u is a given partition of A with }P} ă δ.
Let P1 be the common refinement of P and P1. Write P1 = t∆11, ∆21 , ¨ ¨ ¨ , ∆N1 1u and