Answers & Hints for Selected Exercises 675EXERCISE SET 7.2
n n- Since mi:::; Mi, S,_(f, P) = L mi:::; L Mi= S(f, P).
i=l i=l - Note that P = {a, b} is a partition of [a, b]. For this partition, m 1 = inf 1 [a, b]
and M1 = supl[a,b], so m:::; m1 :::; M1:::; Mand m(b - a):::; m 1 (b-a)=
S..(f, P):::; l:l:::; l:l:::; S(f, P) = M1(b-a):::; M(b-a). - For P1={2,9}, S..(f, P) = m161 = 0 · 7 = 0. :. l:l ~ 0.
Let c > 0. For P2 = {2, 5 - i, 5+ i , 9}, S(f, P) = m161 +m 262 +m 363 =
0 (3 - i) + 1·c+0 (4 - i) = c. .'.Ve> 0, l:l:::; c. By the forcing principle,
1:1:::; o.
Thus we have 0:::; l:l:::; l:l:::; 0. - (a) For P = { 0, ~, 1, ~, 2}, S..(1, P) = 0 · ~ + [ i + ~] ~ + [l + 3] ~ + [ £ + ¥] ~ =
(^2) ; and S(f, P) = [i + ~] ~ + [l + 3H + [£ + ¥J ~ + [4 + 6H = 4;.
(b) For P = { 0, ~, ~, 1, ~, ~, 2}, S..(f, P) = i [ 0 + i + i~ + 4 + ~~] +^2 ; · ~ =
(^438 64) ) and S(l ) P) = l 4 2 [l + 45 16 + 4 + 16 85 + 27] 4 + 10. l 2 = (^294 64).
(c) For P = {o, i, ~' 1, i, 2}, S,(f, P) = i [o + 190 + 292 + 8 + 7i] = 691' and
S(f,P) = i [1 90 + 292 +4+ 1~8+10] = 3N·
(d) For P = { 0, 71 2 4 , 71 , ... ,rt,. 2· .. 2, } -S(f,P) = i;, n [( 2·)2 rt +3 (2 ' )] rt 11 2 =
£ n ~ w [4in2^2 + 21] n = £ n [ 712 4 n(n+l)(2n+l) 6 + .§ n n(n+l) 2 ] = i 3 (1 + .!.) n (2 + .!.)+6 n (1 + .!.) n
i=l
and S,_(f, P) = i~ [ (
2
(i~ll)
2
- 3 (
2
(i~l))] ~ = ~ (1-) (2 - ) +6 (1-*)·
- If {Pn} and { Qn} are sequences of partitions of [a, b] such that S,_(f, Pn) ---+ L
and -S(f, Qn)---+ L, then by (a), (b) and Thm. 7.2.7, L:::; 12:_1:::; rb Jal:::; rb L.
- (a) Let Pn = {2, 2 + ~' 2 + ~" · · , 2 +^3 : }. For each i, Xi = 2 + # and
6i = ~· Since 1 is increasing on [2, 5], mi = l(xi_i) and Mi = l(xi)· Thus,
S(f, Pn) = 'fi Mi6i = i~ (xf - 2xi) · ~ = ~ i~ [ (2 + #)
2- 2 (2 + #) J
= nw l ~ [9i.Ti2^2 + (^21) n ] = l n [ 7126 9 ~ i2 + nw .§_ ~ i] = 113 27 n(n+l){2n+l) 6 + 712 18 n(n+ 2 l )
i=l i=l i=l
= ¥ (1 + ) (2 + ) + 9 (1 + *) ---+ 18.