384 Chapter 7 • The Riemann Integral[ Xi-l, xi] created by Q are equal (to 1). Further, all trapezoidal^6 approxima-
tions and Simpson's rule^7 approximations (if n is even) using the partition
Q are also equal (to 1). But f is not integrable over [O, 1], as was shown in
Example 7.2.10. [See Exercise 21.]EXERCISE SET 7.3- Prove Inequality (2) in the proof of Theorem 7 .3.2. [Use Exercise 7.2.21.]
- Prove Lemma 7.3.4.
- Complete the proof of Part 2 of Theorem 7.3.5 by showing that f: f ::::: I
by the methods used there to prove that f: f :;::: I. - Use the methods of Example 7.3.7 to evaluate each of the following:
(a) fi(2x + 7)dx (b) f 0
4
(4 - 5x)dx - Use the methods of Example 7.3.7 to prove that when 0 <a< b,
(a) f: xdx = ~(b^2 - a^2 ) (b) f: x^2 dx = ~(b^3 - a^3 ) - Prove Theorem 7.3.6 (b).
- Apply the technique of Theorem 7.3.6 to the function f(x) =fa to find
1... { 1 2
2
32
f }0 f usmg the partit10ns Pn = 0, (^2) n , n 2 , (^2) n , · · · ,^1.
- Let f(x) = iji. Use a procedure similar to that used in Exercise 7 to find
fo
1
f.
n- Prove that if f is integrable on [O, 1], then lim .!. L f ( .!£) = f 0
1
f.
n-->oo n k=l n
10. Use the result of Exercise 9, and the integral formulas you learned in
elementary calculus, to evaluate each of the following limits:
n k n k2
(a) lim L 2 (b) lim L 3
n-->oo k=l n n-->oo k=l n
n k5
(c) lim L 6
n-->oo k=l nn 1
(d) lim I: --k
n-->oo k=l n +
n k
(e) lim L 2 k 2
n-->oo k=l n +(f) lim £: n
n-->oo k=l n2 + k2- See Exercise 13.
- See Exercise 14.