7.5 Algebraic Properties of the Integral 3977.5 Algebraic Properties of the Integral
You may have noticed a pattern in earlier chapters. After introducing each of
several big ideas of analysis (limits of sequences, limits of functions, continuous
functions, and derivatives) we included results under the heading "algebra of
[big idea]." This pattern was intentional, to show the similarity and unity of
the concepts and techniques. You may have noticed that even the proofs of
parallel results in the various sections exhibited some similarity. In the present
section we develop similar algebraic results about the Riemann integral.
Theorem 7.5.1 (Algebra of the Integral, I-Linearity) If f and g are
integrable over [a, b] and if c E JR, then
(a) cf is integrable over [a, b], and l: cf= cl: f;
(b) f + g is integrable over [a, b], and l: (! + g) = l: f + l: g.
Proof. (a) Suppose f and g are integrable over [a , b] and c ER Let P*
be any tagged partition of [a, b]. Then
R(cf, P) = cR(f, P).
Hence, in the sense of Theorem 7.3.5,lim R(cf, P*) = lim cR(f, P*) = c lim R(f, P*).
llPll-+O llPll-+O llPll-+OThat is, cf is integrable over [a , b], andl: cf= cl: f.
(b) Similarly, if P * is any tagged partition of [a, b], thenR(f + g, P) = R(f, P) + R(g, P*).
Hence, in the sense of Theorem 7.3.5,lim R(f + g, P*) = lim R(f, P*) + lim R(g, P*).
llPll-+0 llPll-+O llPll-+OThat is, f + g is integrable over [a , b], and l:U + g) = l: f + l: g. •
Theorem 7.5.2 (Algebra of the Integral, II-Preserving Inequalities)
(a) If f is integrable on [a, b] and V'x E [a, b], f(x) ~ 0, then l: f ~ 0.
(b) If f is integrable on [a,b] andV'x E [a,b], m:::; J(x):::; M, thenm(b-a):::;
l: f:::; M(b - a).