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Module MA1S11 (Calculus)

Michaelmas Term 2016

Section 7: Integration

D. R. Wilkins

Copyright

c

David R. Wilkins 2016

Contents

7 Integration 151

7.1 Darboux Sums of a Bounded Function . . . . . . . . . . . . .

151

7.2 Upper and Lower Integrals and Integrability . . . . . . . . . .

155

7.3 Integrability of Monotonic Functions . . . . . . . . . . . . . .

169

7.4 Integrability of Continuous functions . . . . . . . . . . . . . .

172

7.5 The Fundamental Theorem of Calculus . . . . . . . . . . . . .

172

7.6 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . .

178

7.7 Integration by Substitution . . . . . . . . . . . . . . . . . . . .

179

7.8 Indenite Integrals . . . . . . . . . . . . . . . . . . . . . . . .

184

7.9 Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . .

184
i

7 Integration

7.1 Darboux Sums of a Bounded Function

The approach to the theory of integration discussed below was developed by Jean-Gaston Darboux (1842{1917). The integral dened using lower and upper sums in the manner described below is sometimes referred to as the Darboux integralof a function on a given interval. However the class of func- tions that are integrable according to the denitions introduced by Darboux is the class ofRiemann-integrablefunctions. Thus the approach using Dar- boux sums provides a convenient approach to dene and establish the basic properties of theRiemann integral. Letf:[a;b]!Rbe a real-valued function on a closed interval [a;b] that is bounded above and below on the interval [a;b], whereaandbare real numbers satisfyinga < b. Then there exist real numbersmandMsuch that mf(x)Mfor all real numbersxsatisfyingaxb. We seek to dene a quantityRb af(x)dx, thedenite integralof the functionfon the interval [a;b], where the value of this quantity represents the area \below" the graph of the function where the function is positive, minus the area \above" the graph of the function where the function is negative. We now introduce the denition of apartitionof the interval [a;b]. DenitionApartitionPof an interval [a;b] is a setfx0;x1;x2;:::;xngof real numbers satisfying a=x0< x1< x2<< xn1< xn=b: A partitionPof the closed interval [a;b] provides a decomposition of that interval as a union of the subintervals [xi1;xi] fori= 1;2;:::;n, where a=x0< x1< x2<< xn1< xn=b: Successive subintervals of the partition intersect only at their endpoints. LetPbe a partition of the interval [a;b]. ThenP=fx0;x1;x2;:::;xng wherex0;x1;:::;xnare real numbers satisfying a=x0< x1< x2<< xn1< xn=b: The values of the bounded functionf:[a;b]!Rsatisfymf(x)Mfor all real numbersxsatisfyingaxb. It follows that, for each integeri betweeniandn, the set ff(x)jxi1xxig: 151
is a set of real numbers that is bounded below bymand bounded above byM. The Least Upper Bound Principle then ensures that the setff(x)j x i1xxighas a well-dened greatest lower bound and a well-dened least upper bound (see the discussion of least upper bounds and greatest lower bounds in Subsections 1.1.15 to 1.1.19). For each integeribetween 1 andn, let us denote bymithe greatest lower bound on the values of the functionfon the interval [xi1;xi], and let us denote byMithe least upper bound on the values of the functionfon the interval [xi1;xi], so that m i= infff(x)jxi1xxig and M i= supff(x)jxi1xxig: Then the interval [mi;Mi] can be characterized as the smallest closed interval inRthat contains the set ff(x)jxi1xxig: We now consider what the values of the greatest lower bound and least upper bound on the values of the function are determined in particular cases where the function has some special behaviour. First suppose that the functionfis non-decreasing on the interval [xi1;xi]. Thenmi=f(xi1) andMi=f(xi), because in this case the values of the functionfsatisfyf(xi1)f(x)f(xi) for all real numbersxsatisfying x i1xxi. Next suppose that the functionfis non-increasing on the interval [xi1;xi]. Thenmi=f(xi) andMi=f(xi1), because in this case the values of the functionfsatisfyf(xi1)f(x)f(xi) for all real numbersxsatisfying x i1xxi. Next suppose that that the functionfis continuous on the interval [xi1;xi]. The Extreme Value Theorem (Theorem 4.29) then ensures the existence of real numbersuiandvi, wherexi1uixiandxi1vixi with the property that f(ui)f(x)f(vi) for all real numbersxsatisfyingxi1uixi. Thenmi=f(ui) and M i=f(vi). Finally consider the functionf:R!Rdened such thatf(x) =x bxcfor all real numbersx, wherebxcis the greatest integer satisfying the inequalitybxc x. Then 0f(x)<1 for all real numbersx. If the interval [xi1;xi] includes an integer in its interior then supff(x)jxi1xxig= 1; 152
and thusMi= 1, even though there is no real numberxfor whichf(x) = 1. We now summarize the essentials of the discussion so far. The functionf:[a;b]!Ris a bounded function on the closed interval [a;b], whereaandbare real numbers satisfyinga < b. There then exist real numbersmandMsuch thatmf(x)Mfor all real numbersx satisfyingaxb. We are given also a partitionPof the interval [a;b]. This partitionPis representable as a nite set of real numbers in the interval [a;b] that includes the endpoints of the interval. Thus

P=fx0;x1;:::;xng

where a=x0< x1< x2<< xn1< xn=b:

The quantitiesmiandMiare then dened so that

m i= infff(x)jxi1xxig and M i= supff(x)jxi1xxig: fori= 1;2;:::;n. Thenmif(x)Mifor all real numbersxsatisfying x i1xxi. Moreover [mi;Mi] is the smallest closed interval that contains all the values of the functionfon the interval [xi1;xi]. DenitionLetf:[a;b]!Rbe a bounded function dened on a closed bounded interval [a;b], wherea < b, and let the partitionPbe a partition of [a;b] given byP=fx0;x1;:::;xng, where a=x0< x1< x2<< xn1< xn=b: Then thelower sum(orlower Darboux sum)L(P;f) and theupper sum(or upper Darboux sum)U(P;f) offfor the partitionPof [a;b] are dened so that

L(P;f) =nX

i=1m i(xixi1); U(P;f) =nX i=1M i(xixi1); wheremi= infff(x)jxi1xxigandMi= supff(x)jxi1xxig.

ClearlyL(P;f)U(P;f). MoreovernP

i=1(xixi1) =ba, and therefore m(ba)L(P;f)U(P;f)M(ba); for any real numbersmandMsatisfyingmf(x)Mfor allx2[a;b]. 153
RemarkLet us consider how the lower and upper sum of a bounded function f:[a;b]!Ron a closed bounded interval [a;b] are related to the notion of the area \under the graph of the functionf" on the intervala, in the case where the functionfis non-negative on the interval [a;b]. Thus suppose that f(x)0 for allx2[a;b], and letXdenote the region of the plane bounded by the graph of the functionffromx=atox=band the linesx=a, x=bandy= 0. Then

X=f(x;y)2R2jaxband 0yf(x)g;

whereR2is the set of all ordered pairs of real numbers. (The elements ofR2 are then regarded as Cartesian coordinates of points of the plane.)

For each integerilet

X i=f(x;y)2Xjxi1xxig =f(x;y)2R2jxi1xxiand 0yf(x)g: If the regionsXandXihave well-dened areas fori= 1;2;:::;nsatisfying the properties that areas of planar regions are expected to satisfy, then area(X) =nX i=1area(Xi); because, where subregionsXifor dierent values ofiintersect one another, they intersect only along their bounding edges. Letibe an integer between 1 andn. Then 0mif(x) for all real numbersxsatisfyingxi1xxi. It follows that the rectangle with vertices (xi1;0), (xi;0), (xi;mi) and (xi1;mi) is contained in the regionXi. This rectangle has widthxixi1and heightmi, and thus has areami(xixi1).

It follows that

m i(xixi1)area(Xi) for all integersibetween 1 andn. Summing these inequalities overi, we nd that

L(P;f) =nX

i=1m i(xixi1)nX i=1area(Xi) = area(X): An analogous inequality holds for upper sums. For each integeribetween x i1andxithe regionXiof the planeR2is contained within the rectangle with vertices (xi1;0), (xi;0), (xi;Mi) and (xi1;Mi). This rectangle has widthxixi1and heightMi, and thus has areaMi(xixi1). It follows that M i(xixi1)area(Xi) 154
for all integersibetween 1 andn. Summing these inequalities overi, we nd that

U(P;f) =nX

i=1M i(xixi1)nX i=1area(Xi) = area(X): We conclude therefore that if the functionfis non-negative on the interval a;b], and if the regionX\under the graph of the function" on the interval [a;b] has a well-dened area, then

L(P;f)area(X)U(P;f):

7.2 Upper and Lower Integrals and Integrability

DenitionLetfbe a bounded real-valued function on the interval [a;b], wherea < b. Theupper Riemann integralURb af(x)dx(orupper Darboux integral) and thelower Riemann integralLRb af(x)dx(orlower Darboux integral) of the functionfon [a;b] are dened by U Z b a f(x)dx= inffU(P;f)jPis a partition of [a;b]g; L Z b a f(x)dx= supfL(P;f)jPis a partition of [a;b]g: The denition of upper and lower integrals thus requires thatURb af(x)dx be the inmum of the values ofU(P;f) and thatLRb af(x)dxbe the supre- mum of the values ofL(P;f) asPranges over all possible partitions of the interval [a;b]. RemarkLet us consider how the lower and upper Riemann integrals of a bounded functionf:[a;b]!Ron a closed bounded interval [a;b] are related to the notion of the area \under the graph of the functionf" on the intervala, in the case where the functionfis non-negative on the interval [a;b]. Thus suppose that the regionXhas a well-dened area area(X), where

X=f(x;y)2R2jaxband 0yf(x)g:

We have already shown that

L(P;f)area(X)U(P;f)

for all partitionsPof the interval [a;b]. It follows that area(X) is an upper bound on all the lower sums determined by all the partitionsPof [a;b]. It 155
y x x0x1x2x3x4x5x6x7x8The upper sumU(P;f)y x x0x1x2x3x4x5x6x7x8The lower sumL(P;f) 156
is therefore not less than the least upper bound on all these lower sums.

Therefore

LZ b a f(x)dxarea(X);

An analogous argument shows that

U Z b a f(x)dxarea(X): Thus if the regionXhas a well-dened area, then that area must satisfy the inequalities LZ b a f(x)dxarea(X) UZ b a f(x)dx: DenitionA bounded functionf:[a;b]!Ron a closed bounded interval [a;b] is said to beRiemann-integrable(orDarboux-integrable) on [a;b] if U Z b a f(x)dx=LZ b a f(x)dx; in which case theRiemann integralRb af(x)dx(orDarboux integral) offon [a;b] is dened to be the common value ofURb af(x)dxandLRb af(x)dx.

Whena > bwe dene

Z b a f(x)dx=Z a b f(x)dx for all Riemann-integrable functionsfon [b;a]. We setRb af(x)dx= 0 when b=a. Iffandgare bounded Riemann-integrable functions on the interval [a;b], and iff(x)g(x) for allx2[a;b], thenRb af(x)dxRb ag(x)dx, since L(P;f)L(P;g) andU(P;f)U(P;g) for all partitionsPof [a;b]. We recall the basic denitions associated with the denition of the Rie- mann integral (or Riemann-Darboux) integral of a bounded real-valued func- tionf:[a;b]!Ron a closed bounded interval [a;b], whereaandbare real numbers satisfyinga < b. The functionfis required to be bounded, and therefore there exist real numbersmandMwith the property that mf(x)Mfor all real numbersxsatisfyingaxb. ApartitionPof the interval [a;b], may be specied in the formP= fx0;x1;x2;:::;xng, wherex0;x1;:::;xnare real numbers satisfying a=x0< x1< x2<< xn=b 157
The quantitiesmiandMiare dened fori= 1;2;:::;nso that m i= infff(x)jxi1xxig and M i= supff(x)jxi1xxig: Then the interval [mi;Mi] can be characterized as the smallest closed interval inRthat contains the set ff(x)jxi1xxig: TheDarboux lower sumL(P;f) andDarboux upper sumU(P;f) determined by the functionfand the partitionPof the interval [a;b] are then dened by the identities

L(P;f) =nX

i=1m i(xixi1); U(P;f) =nX i=1M i(xixi1):y x x0x1x2x3x4x5x6x7x8Thelower Riemann integralLRb af(x)dxof the functionfon the interval [a;b] is dened to be the least upper bound of the Darboux lower sumsL(P;f) asPranges over all partitions of the interval [a;b].

Similarly theupper Riemann integralLRb

af(x)dxof the functionfon the interval [a;b] is dened to be the greatest lower bound of the Darboux upper sumsU(P;f) asPranges over all partitions of the interval [a;b]. The lower and upper Riemann integrals of the functionfon the intervalquotesdbs_dbs14.pdfusesText_20