7.2 The Riemann Integral Defined 371- Repeat Exercise 6 for the function f(x) = x^2 + 3x.
- Prove Theorem 7.2. 12 (b).
- Prove Theorem 7.2.12 (c).
- Use Exercise 6 and the method of Example 7.2.13 to establish integrability
and find
(a) li(5 - 3x)dx (b) li(x^2 + 3x)dx - Use the method of Example 7.2.13 to establish integrability and find
(a) l:(x^2 - 2x)dx (b) l 1
6
(x^2 + x + 3)dx(c) li x^3 dx (d) 1~ 1 (1 - x^3 )dx
- Suppose f: [a, b] ---+JR is bounded and nonnegative on [a, b]. Prove that
(a) l:f?:.0.
(b) if f is continuous at some xo E (a, b) and f(xo) > 0, then l: f > 0.
[Use the neighborhood inequality property of continuous functions, Ex-
ercise 5.1.26.]
(c) if f is continuous on [a, b], then l: f = 0 ~ \:/x E [a, b], f(x) = 0. To
see what can happen if f is not continuous on [a, b], see Exercise 5.- Suppose f,g: [a,b]---+ JR are bounded [a,b] and \:/x E [a,b], f(x):::; g(x).
Prove that l: f :::; l: g and l: f :S l: g ·
- Suppose f is nonnegative and integrable over [a, b], and f(r) = 0 for all
rational r E [a, b]. Prove that l: f = 0.
- Another Sequential Criterion for Integrability: Suppose {Pn} is
a sequence of partitions of [a, b], each of which is refined by its successor
(i.e., \:/n E N, Pn ~ Pn+i). Prove that for any bounded f : [a, b] ---+JR,
(a) both {~(!, Pn)} and {S(f, Pn)} converge, and n->oo lim ~(!, Pn) :::;
lim S(f, Pn)·
n->oo
(b) if lim ~(!, Pn) = lim S(f, Pn) = L , then f is integrable on [a, b],
n--+oo n--+oo
and l: f = L.
(c) integrability off on [a, b] does not guarantee that lim ~(!, Pn) =
n->OO
lim S(f, Pn)· [Show by counterexample.]
n->oo - Prove Theorem 7.2.14. [Hint: Apply Theorems 7.1.5 and 7.2.4 to the sets
A and B defined in Definition 7.2.6.J