7.2 The Riemann Integral Defined 369andn n b
S(f,P) = _LMd'~i = ,LJ(xi)~. Hence,
i=l i=l n- ~ b-a ~ b-a
S(f, P) - $._(!, P) = L f(xi)-- L f(xi_i)-
i=l n i=l n
b-a~
= - L [f (xi) - f (xi_i)J
n i=lb-a
= - [jkai} - f(a) + ~ -flai} + · · · + f(b) -Wn:::11]
n
(all but two terms cancel out)
b-a
= - [f(b) - f(a)]
n
< (b - a)[f(b) - f(a)]c:.Therefore, by Riemann's criterion for integrability, f is integrable on [a, b].
Case 2 (!is monotone decreasing on [a, b]): Exercise 19. •
Theorem 7.2.17 If f is continuous on [a, b], then f is integrable on
[a,b].
Proof. Suppose f is continuous on [a, b]. Since [a, b] is compact, f is uni-
formly continuous there (see Definition 5.4.l and Theorem 5.4.7).
Let E: > 0. By definition of uniform continuity, :3 8 > 0 3 Vx, y E [a, b],
Ix -YI < 8 ==? lf(x) - f(y)I < E:.
b-a
Choose any integer n > -
8
-, and let P = { xo, X1, x2, · · · , Xn} be thepartition of [a, b] consisting of n equally spaced points. Then each 6i = b - a <
n
- By the extreme value theorem (5.3.7), f assumes minimum and maximum
values on each subinterval [xi-l, xi] created by P. That is, :3 x~, x~' E [xi-l, xi] 3
mi = f(x~) and Mi= J(x~').