7.3 The Integral as a Limit of Riemann Sums 385
11. Modify the techniques used in Exercise 10, as needed, to evaluate each of
the following:
(a) lim ~ f"= cos (k
2
7r)
n-+OO n k=l n
( c) lim ~ 2k + 3n
n-+oo~ n2
k=l
(b) lim -^1 I: n sin (kn) -
n-+oo n k=l 3n
(d) lim ~ fl8k
n-+oo~ V ~
k=l
- Consider the function f(x) = {
1
if
1
:::; x <
3
} ·
0 otherwise
A slight modification of the argument given in Example 7.2.11 will prove
that f is integrable over [O, 4]. Let { Qn} denote the sequence ofregular n-
partitions of [O, 4]. Show that the sequence of lower sums {S..(f, Qn)} is not
monotone increasing, and that the sequence of upper sums {S(f, Qn)} is
not monotone decreasing. (See Remarks 7.3.10.)
13. Prove the trapezoidal rule: If f is integrable over [a, b], and Qn
{xo,x 1 , · · · ,xn} is the regular n-partition of [a,b], then
b b-a
J, a f = n-+oo lim -2n [f (xo) + 2f (x1) + · · · + 2f (xn_i) + f (xn)].
Hint: The expression in brackets is 2 [f(xo) + f(x1) + · · · + f(xn_i) + f(xn)]-
[f(b) + f(a)J.
- Prove Simpson's rule: If f is integrable over [a, b], and Qn = { Xo, x 1 , · · · , Xn}
is the regular n-partition of [a, b] into an even number of subintervals, then
b b-a
J. f = lim - [f (xo) + 4f (x1) + 2f (x2) + 4f (x 3) · · · + 4f (Xn-1) + f (xn)].
a n-+oo 3n
Hint: Try something like the hint given in Exercise 13.
- Finish the proofofTheorem 7.3.11 by showing that S(f, Q) < S(f, P)+c-.
16. Prove Corollary 7.3.12 (b).
- Prove Theorem 7.3.13.
- Prove Theorem 7.3.14.
- Prove Theorem 7.3. 15.
- Prove Theorem 7.3. 16.
- Prove the claims made in Example 7.3.17.
