[PDF] Practice Problems 16 : Integration, Riemanns Criterion for integrability PDF pp16.pdf

(b) Find f : [0,1] → R such that f2 is integrable but f is not integrable 4 Let f and g be two integrable functions on [a, b] (a) If f(x) ≤ g(x) for all x ∈ [a, b], show that

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Suppose that f : [a, b] → R is an integrable function Then f is also integrable on [a, b] Proof Let ϵ > 0 be given Since f is integrable, there exists a partition P

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You may also find that things are clearer if you reorganize the argument a bit ) Solution Suppose f is integrable on [a, b] For Exercise 33 5, we need to show that

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Show that the converse is not true by finding a function f that is not integrable on [ a, b] but that f is integrable on [a, b] Solution 2 Consider the function f(x) = { 1 if x

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Compute the upper and lower Darboux sums associated to the function f(x) 4) Show that if f is integrable on [a, b] and if [c, d] c [a, b] then f is integrable on [c,

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Theorem If f is continuous on [a, b], then f is Riemann-integrable on [a, b] Proof We use the condition (*) to prove that f is Riemann-integrable If ϵ > 0, we set

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the next chapter In this chapter, we define the Riemann integral and prove some Proof If sup g = ∞, then sup f ≤ sup g Otherwise, if f ≤ g and g is bounded

[PDF] Practice Problems 16 : Integration, Riemanns Criterion for integrability

(b) Find f : [0,1] → R such that f2 is integrable but f is not integrable 4 Let f and g be two integrable functions on [a, b] (a) If f(x) ≤ g(x) for all x ∈ [a, b], show that

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If fi [a,b] R is a bounded function, then ffon da e § fonda 2 A bounded If f is Riemann integrable on [a, o and Proof f is continuous on [a, b] = f is bounded

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Suppose there exist sequences (Un) and (Ln) of upper and lower Darboux sums for f such that Un − Ln → 0 Show that f is integrable and that ∫ b a f = lim

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Practice Problems 16 : Integration, Riemann's Criterion for integrability (Part II)

1. Letf: [a;b]!Rbe integrable and [c;d][a;b]. Show thatfis integrable on [c;d].

2. (a) Letfbe bounded on [c;d],M= supff(x) :x2[c;d]g,M0= supfjf(x)j:x2[c;d]g,

m= infff(x) :x2[c;d]gandm0= inffjf(x)j:x2[c;d]g. Show thatM0m0 Mm. (b) Letf: [a;b]!Rbe integrable. Show thatjfjandf2are integrable.

3. (a) Findf: [0;1]!Rsuch thatjfjis integrable butfis not integrable.

(b) Findf: [0;1]!Rsuch thatf2is integrable butfis not integrable.

4. Letfandgbe two integrable functions on [a;b].

(a) Iff(x)g(x) for allx2[a;b], show thatRb af(x)dxRb ag(x)dx. (b) Show thatRb af(x)dxRb ajf(x)jdx. (c) Ifmf(x)Mfor allx2[a;b] show thatm(ba)Rb af(x)dxM(ba):Use this inequality to show thatp3 8 R=3 =4sinxx dxp2 6

5. Letf: [a;b]!Randf(x)0 for allx2[a;b]

(a) Iffis integrable, show thatRb af(x)dx0. (b) Iffcontinuous andRb af(x)dx= 0 show thatf(x) = 0 for allx2[a;b]. (c) Give an example of an integrable functionfon [a;b] such thatf(x)0 for all x2[a;b] andRb af(x)dx= 0 butf(x0)6= 0 for somex02[a;b].

6. Letf: [0;1]!Rbe a bounded function. Suppose that for anyc2(0;1],fis integrable

on [c;1]. (a) Show thatfis integrable on [0;1]. (b) Show that the functionfdened byf(0) = 0 andf(x) = sin(1x ) on (0;1] is integrable,

7. Letf: [a;b]!Rbe a continuous function. Suppose that whenever the productfgis

integrable on [a;b] for some integrable functiong, we haveRb a(fg)(x)dx= 0. Show that f(x) = 0 for everyx2[a;b].

8. (a) Letx;y0. Show that limn!1(xn+yn)1n

=MwhereM= maxfx;yg. (b) Letf: [a;b]!Rbe continuous andf(x)0 for allx2[a;b]. Show that lim n!1Rb af(x)n 1n =MwhereM= supff(x) :x2[a;b]g.

9. (a) (Cauchy-Schwarz inequality) Letx1;x2;:::xn;y1;y2;:::;yn2R. By observing thatPn

i=1(txi+yi)20 for anyt2R, show thatjPn i=1xiyij Pn i=1x2i 12 Pn i=1y2i 12 (b) (Cauchy-Schwarz inequality) Letfandgbe any two integrable functions on [a;b].

Show thatRb

af(x)g(x) 2Rb ajf(x)j2dxRb ajg(x)j2dx

10.(*)Letf: [a;b]!Rbe integrable. Suppose that the values offare changed at a nite

number of points. Show that the modied function is integrable.

11.(*)Letf: [a;b] be a bounded function andE[a;b]. Suppose thatEcan be covered by

a nite number of closed intervals whose total length can be made as small as desired. If fis continuous at every point outsideE, show thatfis integrable.

Practice Problems 16 : Hints/Solutions

1. Let >0. Sincefis integrable on [a;b], there exists a partitionP=fx0;x1;x2;:::;xng

(of [a,b]) such thatU(P;f)L(P;f)< . LetP1=P[ fc;dgandP0=P1\[c;d] which is a partition of [c;d]. Then, sinceMimi>0, it follows thatU(P0;f)L(P0;f) U(P1;f)L(P1;f)U(P;f)L(P;f)< . Apply the Riemann Criterion.

2. (a) Letx;y2[c;d]. Thenjf(x)j jf(y)j jf(x)f(y)j Mm:Fixyand take

supremum forx, we getM0 jf(y)j Mm. Take inmum fory. (b) To show thatjfjis integrable, use the Riemann Criterion and (a). For showingf2is integrable, use the inequality (f(x))2(f(y))22Kjf(x)f(y)j whereK= supfjf(x)j:x2[a;b]gand proceed as in (a).

3. Letf: [0;1]!Rbe dened byf(x) =1 forxrational andf(x) = 1 forxirrational.

Thenjfj=f2. Note thatfis not integrable butjfjis a constant function.

4. (a) Use

Rb ag(x)dxRb af(x)dx=Rb a(gf)(x)dxand Problem 3 of Practice Problems 15 (b) Sincejf(x)j f(x) jf(x)j,x2[a;b], (b) follows from part (a). (c) Use part (a) orL(P;f)Rb af(x)dxU(P;f). On [4 ;3 ],sinxx decreases.

5. (a) This follows from the denition of integrability offor from Problem 4.

(b) Letx02(a;b) be such thatf(x0)> for some >0. Then by the continuity off there exists a >0 such that (x0;x0+)(a;b) andf(x)> on (x0;x0+).

Then we can nd a partitionPof [a;b] such thatRb

af(x)dxL(P;f)> >0. (c) Letf(a) = 1 andf(x) = 0 for allx2(a;b]. ThenRb af(x)dx= 0 butf(a)6= 0.

6. (a) LetM= supfjf(x)j:x2[0;1]g. IfPn=f1n

;x1;x2;:::;xngis a partition of [1n ;1] then letP0n=f0;1n ;x1;x2;:::;xngbe a corresponding partition of [0;1]. ThenU(P0n;f) Mn +U(Pn;f) andL(P0n;f) Mn +L(Pn;f). Therefore,U(P0n;f)L(P0n;f) 2Mn +U(Pn;f)L(Pn;f). For >0, rst choosensuch that2Mn <2 and then choosePnsuch thatU(Pn;f)L(Pn;f)<2 . Apply the Riemann Criterion. (b) Sincefis continuous on [c;1] for everycsatisfying 0< c <1,fis integrable on [c;1].

Apply part (a).

7. Supposef(x0)>0 for somex02(a;b). Use the argument used in Problem 5(b).

8. (a) Note thatM(xn+yn)1n

(2Mn)1n . Use the Sandwich Theorem. (b) For >0, by the continuity off,9[c;d][a;b] such thatf(x)> M8x2[c;d].

Hence (M)(dc)1n

Rb af(x)n 1n

M(ba)1n

. Apply the Sandwich Theorem.

9. We will see the solution of part (b) and the solution of part (a) is similar. Note that the

inequalityRb a(tf(x)g(x))2=t2Rb af2(x)dx 2tRb af(x)g(x)dx +Rb ag2(x)dx 0 holds for allt2R. Taket= where=Rb af(x)g(x)dxand=Rb af2(x)dx.

10. Suppose the values offare changed atc1;c2;::;cpandgis the modied function. Let

M= maxfjg(c1)j;jg(c2)j;:::;jg(cp)jg. Let >0. Sincefis integrable, there exists a partitionPof [a;b] such thatU(P;f)L(P;f)<2 . Coverc0isby the intervals [y1;y2];[y3;y4];:::;[y2p1;y2p] wherey0isare in [a;b] andjy1y2j+jy3y4j+:::+jy2p1 y

[PDF] [PDF] Practice Problems 16 : Integration, Riemanns Criterion for integrability