THE FUNDAMENTAL THEOREM OF THE CALCULUS - THE CALCULUS - What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

CHAPTER VIII. THE CALCULUS

§5. THE FUNDAMENTAL THEOREM OF THE CALCULUS

1. The Fundamental Theorem

The notion of integration, and to some extent that of differentiation, had been fairly well developed before the work of Newton and Leibniz. To start the tremendous evolution of the new mathematical analysis but one more simple discovery was needed. The two apparently unconnected limiting processes involved in the differentiation and integration of a function are intimately related. They are, in fact, inverse to one another, like the operations of addition and subtraction, or multiplication and division. There is no separate differential calculus and integral calculus, but only one calculus.

It was the great achievement of Leibniz and Newton to have first clearly recognized and exploited this fundamental theorem of the calculus. Of course, their discovery lay on the straight path of scientific development and it is only natural that several men should have arrived at a clear understanding of the situation independently and at almost the same time.

To formulate the fundamental theorem we consider the integral of a function y = f(x) from the fixed lower limit a to the variable upper limit x. To avoid confusion between the upper limit of integration x and the variable x that appears in the symbol f(x), we write this integral in the form (see p. 412)

(1) image

indicating that we wish to study the integral as a function F(x) of the upper limit x (Fig. 274). This function F(x) is the area under the curve y = f(u) from the point u = a to the point u = x. Sometimes the integral F(x) with a variable upper limit is called an “indefinite” integral.

image

Fig. 274. The integral as function of upper limit.

Now the fundamental theorem of the calculus is:

The derivative of the indefinite integral (1) as a function of x is equal to the value of f(u) at the point x:

F′(x) = f(x

In other words, the process of integration, leading from the function f(x) to F(x), is undone, inverted, by the process of differentiation, applied to F(x).

On an intuitive basis the proof is very easy. It depends on the interpretation of the integral F(x) as an area, and would be obscured if one tried to represent F(x) by a graph and the derivative F′(x) by its slope. Instead of this original geometrical interpretation of the derivative we retain the geometrical explanation of the integral F(x) but proceed in an analytical way with the differentiation of F(x). The difference

F(x1) – F(x)

is simply the area between x and x1 in Figure 275, and we see that this

image

Fig. 275. Proof of the fundamental theorem.

area lies between the values (x1x)m and (x1x)M,

(x1x)mF(x1) – F(x) ≤ (x1x)M,

where M and m are respectively the greatest and least values of f(u) in the interval between x and x1. For these two products are the areas of rectangles including the curved area and included in it, respectively. Therefore

image

We shall assume that the function f(u) is continuous, so that if x1 approaches x, then M and m both approach f(x). Hence we have

(2) image

as stated. Intuitively, this expresses the fact that the rate of change of the area under the curve y = f(x) as x increases is equal to the height of the curve at the point x.

In certain textbooks the salient point in the fundamental theorem is obscured by poorly chosen nomenclature. Many authors first introduce the derivative and then define the “indefinite integral” simply as the inverse of the derivative, saying that G(x) is an indefinite integral of f(x) if

G′(x) = f(x).

Thus their procedure immediately combines differentiation with the word “integral.” Only later is the notion of the “definite integral” as an area or as the limit of a sum introduced, without emphasizing that the word “integral” now means something totally different. In this way the main fact of the theory is smuggled in by the back door, and the student is seriously impeded in his efforts to attain real understanding. We prefer to call functions G(x) for which G′(x) = f(x) not “indefinite integrals” but primitive functions of f(x). The fundamental theorem then simply states:

F(x), the integral of f(u) with fixed lower limit and a variable upper limit x, is a primitive function of f(x).

We say “a” primitive function and not “the” primitive function, for it is immediately clear that if G(x) is a primitive function of f(x), then

H(x) = G(x) + c  (c any constant)

is also a primitive function, since H′(x) = G′(x). The converse is also true. Two primitive functions, G(x) and H(x), can differ only by a constant. For the difference U(x) = G(x) – H(x) has the derivative U′(x) = G′(x) – H′(x) = f(x) − f(x) = 0, and is therefore constant, since a function represented by an everywhere horizontal graph must be constant.

This leads to a most important rule for finding the value of an integral between a and b, provided we know a primitive function G(x) of J(x). According to our main theorem,

image

is also a primitive function of f(x). Hence F(x) = G(x) + c, where c is a constant. The constant c is determined if we remember that F(a) = image. This gives 0 = G(a) + c, so that c = –G(a). Then the definite integral between the limits a and x will be F(x) = image, or, if we write b instead of x,

(3) image

irrespective of what particular primitive function G(x) we have chosen. In other words,

To evaluate the definite integral image we need only find a function G(x) such that G′(x) = f(x), and then form the difference G(b) - G(a).

2. First Applications. Integration of xr, cos x, sin x. Arc tan x

It is not possible here to give an adequate idea of the scope of the fundamental theorem, but the following illustrations may give some indication. In actual problems encountered in mechanics, physics, or pure mathematics, it is very often a definite integral whose value is wanted. The direct attempt to find the integral as the limit of a sum may be difficult. On the other hand, as we saw in §3, it is comparatively easy to perform any kind of differentiation and to accumulate a great wealth of information in this field. Each differentiation formula, G′(x) = f(x), can be read inversely as providing a primitive function G(x) for f(x). By means of the formula (3), this can be exploited for calculating the integral of f(x) between any two limits.

For example, if we want to find the integral of x2 or x3 or xn we can now proceed much more simply than in §1. We know from our differentiation formula for xn that the derivative of xn is nxn–1, so that the derivative of

image

is

image

Therefore xn+1/(n + 1) is a primitive function of f(x) = xn, and hence we have immediately

image

This process is much simpler than the laborious procedure of finding the integral directly as the limit of a sum.

More generally, we found in §3 that for any rational s, positive or negative, the function xs has the derivative sxs-1, and therefore, for s = r + 1, the function

image

has the derivative f(x) = G′(x) = xr. (We assume r ≠ – 1, i.e. s ≠ 0.) Hence xr+1/(r + 1) is a primitive function or “indefinite integral” of xr, and we have (for a, b positive and r ≠ – 1)

(4) image

In (4) we suppose that in the interval of integration the integrand xr is defined and continuous, which excludes x = 0 if r < 0. We therefore make the assumption that in this case a and b are positive.

For G(x) = –cos x we have G′(x) = sin x, hence

image

Likewise, since for G(x) = sin x we have G′(x) = cos x, it follows that

image

A particularly interesting result is obtained from the formula for the differentiation of the inverse tangent, D arc tan x = 1/(1 + x2). It follows that the function arc tan x is a primitive function of 1/(1 +x2), and we obtain from formula (3) the result

image

Now we have arc tan 0 = 0 because to the value 0 of the tangent the value 0 of the angle is attached. Hence we find

(5) image

If in particular b = 1, then arc tan b will be equal to π/4, because to the value 1 of the tangent corresponds an angle of 45°, or in radian measure π/4. Thus we obtain the remarkable formula

(6) image

This shows that the area under the graph of the function y = 1/(1 + x2) from x = 0 to x = 1 is one-fourth of the area of a circle of radius 1.

image

Fig. 276 π/4 as area under y = 1/(1 + x2) from 0 to 1.

3. Leibniz’ Formula for π

The last result leads to one of the most beautiful mathematical discoveries of the seventeenth century—Leibniz’ alternating series for π,

(7) image

By the symbol + · · · we mean that the sequence of finite “partial sums”, formed by breaking off the expression on the right after n terms, converges to the limit π/4 as n increases.

To prove this famous formula, we have only to recall the finite geometrical series image, or

image

In this algebraic identity we substitute q = – x2 and obtain

(8) image

where the “remainder” Rn is

image

Equation (8) can now be integrated between the limits 0 and 1. By rule (a) of §3, we have to take on the right the sum of the integrals of the single terms. Since, by (4), image, we find image, and therefore

(9) image

where image. According to (5), the left side of (9) is equal to π/4. The difference between π/4 and the partial sum

image

is π/4 – Sn = Tn. What remains is to show that Tn approaches zero as n increases. Now

image

Recalling formula (13) of §1, which states that image, we see that

image

since the right side is equal to 1/(2n + 1), as we saw before (formula (4)), we find | Tn > 1/(2n + 1). Hence

image

But this shows that Sn tends with increasing n to π/4, since 1/(2n + 1) tends to zero. Thus Leibniz’ formula is proved.