Chapter 7

The IntegralThe traditional treatment of the integral is described in the problems beginning with Prob-lem 25 on Page 95. This approach is due to Darboux and is his description of the Riemannintegral. The Riemann integral dates from the 1850’s. It includes the case of continuousand piecewise continuous functions. However, the first integral to be adequate for consid-ering continuous functions was due to Cauchy in 1820’s who gave the first correct proofof the fundamental theorem of calculus which is normally credited to Newton and Liebniz.This is because the concept of what was meant by the integral was not precisely describeduntil Cauchy. In fact, Cauchy’s integral involved one sided sums and only worked wellon continuous functions, but it was sufficient to give an acceptable proof for the funda-mental theorem of calculus. For a complete discussion including Stieltjes integrals, a veryimportant generalization, see my single variable advanced calculus book on my web site.

Although their understanding of the integral was incomplete, they found it as follows:

Procedure 7.0.1 To find∫ b

a f (x)dx do the following:

1. Find F (x) such that F ′ (x) = f (x) . Such an F is called an anti-derivative. Its deriva-tive is an appropriate one sided derivative at the end points.

2. Then∫ b

a f (x)dx ≡ F (b)−F (a)

The above procedure to find “∫ b

a f (x)dx”, was called the fundamental theorem of cal-culus. What they thought they were getting was a kind of infinite sum of the quantitiesf (x)dx,dx being an “infinitesimal” change in x, which is why it is denoted as

∫ ba f (x)dx,

the long S symbolizing sum. Until Cauchy it was like this: We have something which wedon’t understand but it is like a sum and we can find it by using antiderivatives. It was likereligious ritual. By contrast, Cauchy said exactly what he meant by the integral and showedthat you could find it using antiderivatives. This is a big improvement.

However, defining the integral by the above Procedure, this is in fact a very inter-esting concept, because many applications can be formulated directly as a solution to aninitial value problem from differential equations. This is a problem of the form F ′ (x) =f (x) ,F (0) = F0 where F is an unknown function and F (0) = F0 is an initial condition.Here is a simple example, which is also the main historical motivaton for the integral.

Consider A(x) the area under the graph of a curve y = f (x) as shown in the followingpicture between a and x.

aA(x)

y = f (x)

x x+h

A(x+h)−A(x)

Thus A(x+h)−A(x) ∈ [ f (x)h, f (x+h)h] and so

A(x+h)−A(x)h

∈ [ f (x) , f (x+h)]

Then taking a limit as h → 0, one obtains A′ (x) = f (x) ,A(a) = 0 and so one would have,from the above definition of the integral in terms of a procedure, A(x) =

∫ xa f (t)dt. This

suggests that we should define the area under the graph of the curve between a and x > a

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