400 Chapter 7 • The Riemann Integral(c) Finally, putting together the results of (16) and (19), we have
n
I; (Mi(g of) - mi(g of)) £::,i < c(b - a)+ (B - A)c
i=l
= [(b - a)+ (B - A)]c.Hence, S(g of, P) - $._(go f, P) < [(b -a)+ (B - A)Jc.
Therefore, by Riemann's criterion for integrability, go f is integrable on
[a,b]. •Corollary 7.5.5 (Algebra of the Integral, IV-Absolute Value) If f is
integrable on [a, b], then so is lfl. Moreover, Jl: Jj :::; l: Iii :::; M(b-a), where
Mis any upper bound for lfl on [a,b].Proof. Suppose f is integrable on [a, b]. Then it is bounded there, so
3M> 0 3 Vx E [a,b], lf(x)I :::; M. Let g(x) = lxl. Since g is continuous
everywhere, we can apply Theorem 7.5.4 to conclude that lfl is integrable on
[a, b]. By Theorem 7.5.1, so is -lfl, and l: -Iii = -l: lfl. Now, Vx E [a, b],
-M:::; -lf(x)I:::; J(x):::; lf(x)I :::; M.
Thus, by Theorems 7.5.2 (d) and 7.2.9,- M(b - a)= l: -M:::; - l: Iii :::; l: f:::; l: Iii :::; l: M = M(b - a)
i.e., [l: f [ :::; l: Iii :::; M(b - a). •
Corollary 7.5.6 (Algebra of the Integral, V-Miscellany) If f is inte-
grable on [a, b], then
(a) Vn EN, r is integrable on [a,b].
(b) If f is positive and bounded away from 0 on [a, b], then l/ f is integrable
on [a,b].(c) For n EN, if (j exists Vx E [a, b], then (j is integrable on [a, b].
(d) sinf(x), cosf(x), and ef(x) are all integrable on [a,b].(e) If f is positive and bounded away from 0 on [a,b], then Inf is integrable
on [a,b].Proof. Exercise 8. •