7.5 Algebraic Properties of the Integral 401Corollary 7.5.7 (Algebra of the Integral, VI-Products and Max/Min)
If f and g are integrable on [a, b], then
(a) f g is integrable on [a, b],(b) max{f,g} is integrable on [a,b], and(c) min{f,g} is integrable on [a,b].1
Proof. To prove (a), show that Jg = 4: [(f + g)^2 - (f - g)^2 ] and thenapply Theorem 7.5.l and Corollary 7.5.6 (a). To prove (b) and (c), recall from
Exercise 1.2-B.6 that
max{f, g} = f + g ~If-gl and min{f,g} = f + g ~If-gl
and apply Corollaries 7.5.5 and 7.5.6. See Exercise 12. •
EXERCISE SET 7.5- Suppose f, g : [a, b] ---t IR are bounded on [a, b].
(a) Prove that J:(f + g) ~ J:f + J:g.
(b) Find functions f , g for which strict inequality holds in (a).(c) State and prove similar results for J:(f + g).
(d) Use (a) and (c) to give an alternate proof of Theorem 7.5.1 (c).- Suppose Ji,f2,··· ,Jn are all integrable on [a,b], and c1,c2,· · · ,en E IR.
rb n n rb
Prove that i~ cdi is integrable on [a, b], and } a 8 Cifi = 8 Ci } a k
[Use mathematical induction.]- A function f:[-a, a] ---t IR is an even function if 'ix E [-a, a], f(-x) =
f(x), and is an odd function if 'ix E [-a, a], f(-x) = - f(x). Suppose
f is integrable on [-a, a]. Prove that
(a) if f is even, then f~a f = 2 f 0 a f.
(b) if f is odd, then f~a f = 0.
- Find a function f:[O, 1] ---t IR such that f^2 is integrable on [O, 1] but f is
not. - Prove Theorem 7.5.2.