Definite Integrals
11.1 The Riemann Integral
Definite Integrals
In the introduction to Chap. 10 the notion of the definite integral of a functionf on an interval[a, b]has already been mentioned. It arises from summing up expres-sions of the formf (x)xand taking limits. Such sums appear in many applications including the calculation of areas, surface areas and volumes as well as the calcula-tion of lengths of curves. This chapter employs the nocalcula-tion of Riemann integrals as the basic concept of definite integration. Riemann’s approach provides an intuitive concept in many applications, as will be elaborated in examples at the end of the chapter.
The main part of Chap.11is dedicated to the properties of the integral. In partic-ular, the two fundamental theorems of calculus are proven. The first theorem allows one to calculate a definite integral from the knowledge of an antiderivative. The second fundamental theorem states that the definite integral of a functionf on an interval[a, x]with variable upper bound provides an antiderivative off. Since the definite integral can be approximated, for example by Riemann sums, the second fundamental theorem offers a possibility to approximate the antiderivative numeri-cally. This is of importance, for example, for the calculation of distribution functions in statistics.
11.1 The Riemann Integral
Example 11.1 (From velocity to distance) How can one calculate the distancew which a vehicle travels between timea and timebif one only knows its velocity v(t )for all timesa≤t≤b? Ifv(t )≡vis constant, one simply gets
w=v·(b−a).
If the velocityv(t )is time-dependent, one divides the time axis into smaller subin-tervals (Fig.11.1):a=t0< t1< t2<· · ·< tn=b.
Choosing intermediate pointsτj∈ [tj−1, tj]one obtains approximately v(t )≈v(τj) fort∈ [tj−1, tj],
M. Oberguggenberger, A. Ostermann, Analysis for Computer Scientists, Undergraduate Topics in Computer Science,
DOI10.1007/978-0-85729-446-3_11, © Springer-Verlag London Limited 2011
135
136 11 Definite Integrals Fig. 11.1 Subdivision of the
time axis
Fig. 11.2 Sums of rectangles as approximation to the area
if v is a continuous function of time. The approximation is the more precise, the shorter the intervals[tj−1, tj]are chosen. The distance travelled in this interval is approximately equal to
wj≈v(τj)(tj−tj−1).
The total distance covered between timeaand timebis then w=
n j=1
wj≈ n j=1
v(τj)(tj−tj−1).
Letting the length of the subintervals[tj−1, tj]tend to zero, one expects to obtain the actual value of the distance in the limit.
Example 11.2 (Area under the graph of a non-negative) In a similar way one can try to approximate the area under the graph of a functiony=f (x)by using rectangles which are successively refined (Fig.11.2).
The sum of the areas of the rectangles F ≈
n j=1
f (ξj)(xj−xj−1)
form an approximation to the actual area under the graph.
The two examples are based on the same concept, the Riemann integral,1which we will now introduce. Let an interval [a, b] and a function f = [a, b] →R be given. Choosing the points
a=x0< x1< x2<· · ·< xn−1< xn=b,
1B. Riemann, 1826–1866.
11.1 The Riemann Integral 137 the intervals [x0, x1], [x1, x2], . . . ,[xn−1, xn] form a partition Z of the interval [a, b]. We denote the length of the largest subinterval byΦ(Z), i.e.,
Φ(Z)= max
j=1,...,n|xj−xj−1|.
For arbitrarily chosen intermediate pointsξj∈ [xj−1, xj]one calls the expression S=
n j=1
f (ξj)(xj−xj−1)
a Riemann sum. In order to further specify the idea of the limiting process above, we take a sequenceZ1, Z2, Z3, . . .of partitions such thatΦ(ZN)→0 asN → ∞ and corresponding Riemann sumsSN.
Definition 11.3 A functionf is called Riemann integrable in[a, b]if, for arbitrary sequences of partitions (ZN)N≥1 withΦ(ZN)→0, the corresponding Riemann sums(SN)N≥1tend to the same limitI (f ), independently of the choice of the in-termediate points. This limit
I (f )= b
a
f (x)dx
is called the definite integral off on[a, b].
The intuitive approach in the introductory Examples11.1and11.2can now be made precise. If the respective functionsf andvare Riemann integrable, then the integral
F = b
a
f (x)dx
represents the area between thex-axis and the graph, and w=
b
a
v(t )dt
gives the total distance covered.
Experiment 11.4 Open the M-filemat11_1.m, study the given explanations and experiment with randomly chosen Riemann sums for the functionf (x)=3x2 in the interval[0,1]. What happens if you take more and more partition pointsn?
Experiment 11.5 Open the applet Riemann sums and study the effects of changing the partition. In particular, vary the maximum length of the subintervals and the choice of intermediate points. How does the sign of the function influence the result?
138 11 Definite Integrals
The following examples illustrate the notion of Riemann integrability.
Example 11.6 (a) Letf (x)=c=constant. Then the area under the graph of the function is the area of the rectanglec(b−a). On the other hand, any Riemann sum is of the form
f (ξ1)(x1−x0)+f (ξ2)(x2−x1)+ · · · +f (ξn)(xn−xn−1)
=c(x1−x0+x2−x1+ · · · +xn−xn−1)
=c(xn−x0)=c(b−a).
All Riemann sums are equal and thus, as expected, b
a
cdx=c(b−a).
(b) Letf (x)=x1forx∈(0,1], f (0)=0. This function is not integrable in[0,1]. The corresponding Riemann sums are of the form
1
ξ1(x1−0)+ 1
ξ2(x2−x1)+ · · · + 1
ξn(xn−xn−1).
By choosingξ1close to 0 every such Riemann sum can be made arbitrarily large, thus the limit of the Riemann sums does not exist.
(c) Dirichlet’s function2 f (x)=
1, x∈Q 0, x ∈Q
is not integrable in[0,1]. The Riemann sums are of the form SN=f (ξ1)(x1−x0)+ · · · +f (ξn)(xn−xn−1).
If allξj∈QthenSN=1. If one takes allξj ∈QthenSN=0, thus the limit depends on the choice of intermediate pointsξj.
Remark 11.7 Riemann integrable functionsf: [a, b] →Rare necessarily bounded.
This fact can easily be shown by generalising the argument in Example11.6(b).
The most important criteria for Riemann integrability are outlined in the follow-ing proposition. Its proof is simple, however, requires a few technical considera-tions about refining particonsidera-tions. For details, we refer to the literature, for instance [4, Chap. 5.1].
2P.G.L. Dirichlet, 1805–1859.
11.1 The Riemann Integral 139 Fig. 11.3 A piecewise
continuous function
Proposition 11.8 (a) Every function which is bounded and monotonically increas-ing (monotonically decreasincreas-ing) on an interval[a, b]is Riemann integrable.
(b) Every piecewise continuous function on an interval[a, b]is Riemann inte-grable.
A function is called piecewise continuous if it is continuous except for a finite number of points. At these points, the graph may have jumps but is required to have left- and right-hand limits (Fig.11.3).
Remark 11.9 By taking equidistant grid pointsa=x0< x1<· · ·< xn−1< xn=b for the partition, i.e.,
xj−xj−1=:x=b−a n , the Riemann sums can be written as
SN= n j=1
f (ξj)x.
The transition x→0 with simultaneous increase of the number of summands suggests the notation
b
a
f (x)dx.
Originally it was introduced by Leibniz3with the interpretation as an infinite sum of infinitely small rectangles of width dx. After centuries of dispute, this interpretation can be rigorously justified today within the framework of nonstandard analysis (see for instance [25]).
Note that the integration variablex in the definite integral is a bound variable and can be replaced by any other letter:
b
a
f (x)dx= b
a
f (t )dt= b
a
f (ξ )dξ= · · ·.
3G. Leibniz, 1646–1716.
140 11 Definite Integrals This can be used with advantage in order to avoid possible confusion with other bound variables.
Proposition 11.10 (Properties of the definite integral) In the following, leta < b andf, gbe Riemann integrable on[a, b].
(c) Sum and constant factor (linearity):
b
then one obtains the validity of the sum formula even for arbitrarya, b, c∈Rif f is integrable on the respective intervals.