678 Appendix C • Answers & Hints for Selected Exercises
EXERCISE SET 7 .3- Using the notation of Exercise 7.2.21,
S..(f, Q) - 5._(f, P) = mk,1(xj. -Xk-1) + mk,2(xk -xj.) - mk(Xk -Xk-1)
:::; M(xj.-Xk-1)+M(xk-xj.)+M(xj.-Xk-1)+M(xk-xj.):::; 4M llQll-- Replace the appropriate section of the proof by the following:
By the c-criterion for infimum, we can select tags x; E [xi-1, xi] 3 f (xi) >
n n
Mi-b:_a. Then R(f, P*) = L f(xi)6i > L (Mi-b:_a)6i = S(f, P)-c;. Thus,
i=l i=l
S(f, P) < R(f, P*) + c <I+ 2c. .-. \:/c; > 0, S(f, P) < I+ 2c, so l: f < I+ 2c.
Therefore, by the forcing principle, I 2: l: f.
- Each given function is continuous on the given interval, so it is integrable
there. Let Pn = { xo, x 1 ,. · · , Xn} be the partition of [a, b] into n subintervals
of equal length, 6 = b:;:;a. Then Xi = a+ i6; choose x; = Xi· Then llPn// =
b:;:;a -+ 0, so by Thro. 7.3.6, R(f, Pn) -+ l: f ·
3 n
(d) J_ 1 (x^3 +2x)dx; 6 =~'Xi= -1 +~-Then R(f, Pn) = L j(xi)6 =
i=liti [ ( -1 + ~) 3 + 2 ( -1 + ~) J. ~ = ~ [-3n + ~ iti i - ~ iti i2 + ~ iti i3]
-12 + 40 (1 + ~) - 32 (1 + ~) (2 + ~) + 64 (1 + ~)
2
-+ 28.n- (a) Using the notation used in solving Ex. 4, R(f, Pn) = L f(xi)6 =
i=l
iti(a+i6)6 = b:;:;a iti (a+i (b:;:;a)) = b:;:;a [an+ b:;:;an(n2+1)]
= (b - a)a + (b-a)2 2 (1 + n l) -+ ab - a2 + ~ 2 - ab+ a2 2-- b2-a2 2 ·
L _ {^1(^2 22 32) n (^2) } i (^2) (i-1) (^2) 2i-1 2
- et Pn - 0, 11:2> 11:2> 11:2> · • · , 11:2 • Then 6i - 11:2 - ~ - --ri2 < n' so
llPnll < ~-+ 0. Let x; =Xi· Then R(f, Pn) = £= f(xi)6i = £= [¥; ·^2 ~2^1
i=l i=l
n
= '\""' 2i^2 -i = 2 n(n+1)(2n+l) - 1 n(n+l) -+ ±.. rl f = ±.
L., ---;:;;r 7t3 6 7t3 2 3 ... J 0 3.
i=l
- Suppose f is integrable on [a, b], and let Pn = {O, ~' ~' · · · , ~}.Then Xi={:;
and 6i =~-To make Pa tagged partition, choose x; =Xi· Then R(f, P~) =
n n. n. 1
L f(xi)6i = L JC!;)~ = ~ L !(*). By Thro. 7.3.6, lim R(f, P~) = lo f
i=l i=l i=l n--+oo
since llPnll = l-+ 0. Thus, lim l £= J(.i) = l~ f.
n n-+cx:> n i=l n