Answers & Hints for Selected Exercises 679- (a) 1/ 2 (b) Let f(x) = x^2 • Then
n k2 n 2 n 1 1
n-+lim oo iL =l 7i3 = n-lim Hx:i ~ i=l L (~) = n-+oo lim ~ iL =l f (~) = f^0 f = f^0 x^2 dx = ~·
(c) 1/ 6 (d) ln2 (e) ~ ln2 (f) 7r/4 - (a) Think of f ( x) = cos x over [ 0, ~], with P n dividing the interval
into n subintervals of equal length. Then Vi, Xi = ~~, and llPnll = 2 : ---+
n n - Choose xi = Xi · Then R(f, P~) = I: f(x i)L.i = L cos(~~) 2 :. Thus,
i=l i=l
n-+lim oo .!. n if =l cos ( ~7r) n = 7r ±. nlim -+oo R(f, P~) = 1r. ±. f^0 1r^12 cos xdx = ±-7r.
(b) 3/(27r) (c) 4 (d) v's- Let Pn = {x o, X 1 , · · · , Xn} be the partition of [a, b] into n subintervals of
equal length, 6 = b:;;_a; let P~ be the tagged partition using the left endpoints
of each subinterval [xi-l, x i], and let P~ be the t agged partition using the
right endpoints. By Thm. 7.3.6, t a f = n-+lim oo R(f, P~) = n-+lim oo R(f, P~). Thus,
J: f = ~ [ nlim -+oo R(f, P~) + n-+lim oo R(f, P~)] = ~ nlim -+oo [R(f, P~) + R(f, P~)] =
~ nl!_.1! [ b:;;_a i~ f(Xi-1) + b:;;_a i~ f(x i)]
= lim b;: [ f ( xo) + 2 L f ( xi ) + f ( xn) ·
n-1 ]
n-+oo i=l- Revise the portion of the proof beginning in line 7, as follows: change m i,
mk, inf, and::; to Mi, Mk, sup, and:'.'.'.. Also, change "+B" and "+(B-mk 1 )"
to "-B" and "-(B + MkJ", and so on. - (::::}) Suppose f integrable on [a,b]. By Cor. 7.3.12, :3 L E JR 3 L =
sup{~(!, Q) : Q is a regular partition of [a, b]} = inf{S(f, Q) : Q is a regular
partition of [a, b]}. Let c > 0. By the €-criterion for sup and inf, :3 regular
partitions Qm, Qn of [a, b] 3 ~(!, Qm) > L - ~ and S(f, Qn) < L + ~· For
k :'.'.'. max{m, n}, S(f, Qk) - ~(!, Qk)::; S(f, Qn) - ~(!, Qm) < E:.
(-{::::) Apply Thm. 7.2.14.- (::::}) Suppose f integrable on [a,b]. Let c > 0. By Thm. 7.3.2, :3 8 > 0 3 V
partitions P of [a, b], llPll < 8 ::::} S(f, P) - ~(!, P) < c. Choose no E N 3
b;;, 0 a < 8. Then n :'.'.'.no::::} llQnll = b:;;_a < 8::::} S(f, Qn) - ~(!, Qn) < c.
(-{::::) Apply Riemann's criterion, Thm. 7.2.14.- Let f be the Dirichlet function on [O, 1], and Qn = { 0, ~, ~, · · · , ~}. Note
that Xi, Xi-l , and x;--+x;-i^11 ·^1 Th
2 - are a rat10na. us ,