Elements of Mathematics: From Euclid to Gödel by John Stillwell

Elements of Mathematics: From Euclid to Gödel by John Stillwell

Author:John Stillwell [Stillwell, John]
Language: eng
Format: epub
Publisher: Princeton University Press
Published: 2016-05-31T05:00:00+00:00


214

Chapter 6

A very similar argument, using the fact that d

dx xn+1 = ( n + 1) xn,

gives:

Area under y = tn. For any positive integer n, the area under y = tn between t = 0 and t = x is well-defined and equals 1

n+ xn+1 .

1

It is possible to avoid using the hard zero derivative theorem in

finding these areas, but only at the cost of considerable algebra to find exact formulas for the sums of upper and lower rectangles. This gets

harder and harder as n increases, and in any case it turns a blind eye to the really important insight of these proofs: finding area is in some sense inverse to finding derivatives. This insight is worth elaborating, because it leads to a fundamental theorem.

6.6 ∗The Fundamental Theorem of Calculus

The idea of area under a curve is captured in calculus by the concept of integral. There are several concepts of integral, but elementary calculus deals only with the simplest one, the Riemann integral, and applies it only to continuous functions.

Given a function y = f ( t), continuous from t = a to t = b, the integral of f from a to b is written

b f( t) dt,

a

and it is defined in the same way that we defined “area under the curve y = f ( t)” for certain functions f in the previous section. Namely, we divide the interval [ a, b] into finitely many parts and approximate the graph of f from above and below by rectangles erected on these parts

(figure 6.11).

If it is possible to make the difference between the upper and lower

approximations arbitrarily small, then there is a single number that lies between them (by the completeness of R), and this number is the value

b

of a f ( t) dt.

It is very plausible that, for a continuous function, the difference

between upper and lower approximations can be made arbitrarily small.

But proving this is a delicate matter, much like the proof of the zero

Calculus

215

y

y = f( t)

t

O

a

b

b

Figure 6.11: Approximating

f

a

( t) dt by rectangles.

derivative theorem in section 6.3. (In fact, the heart of the proof is an infinite bisection process, like the one used there.) For this reason, proving the existence of the Riemann integral for continuous functions belongs to advanced calculus.

However, if we make the plausible assumption that the integral

exists, we can proceed to differentiate this integral, just as we differentiated the special area functions in the previous section. This gives: Fundamental theorem of calculus. If f is continuous on an interval

[ a, b] , and

x

F ( x) =

f ( t) dt,

a

then F ( x) = f ( x) .

The fundamental theorem can be used to identify functions F ( x) defined by integrals in the case where we know a function G( x) whose derivative is f ( x) (just as we did for the area functions in the previous section). In this case, the zero derivative theorem tells us that F ( x) differs from G( x) only by a constant.

The fundamental theorem is



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