Primitive, Definite Integral, Fundamental Theorem of Integral Calculus, Properties of Definite Integrals

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PRIMITIVE, DEFINITE INTEGRAL, FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS, PROPERTIES OF DEFINITE INTEGRALS

Def. Primitive of a function. Given a function f(x), the primitive of f(x) is a function F(x) which has f(x) as its derivative i.e. it is a function F(x) such that F'(x) = f(x). If F(x) is a primitive of f(x) then F(x) + c is also a primitive of f(x), where c is any constant. Thus if a function f(x) has one primitive F(x), then along with this one it has an entire family of primitives, infinite in number, of the form F(x) + c. And this family of primitives exhausts the whole set of primitives for f(x) — there are no others.

Syn. Anti-derivative, indefinite integral.

Example. Find the primitive of the function 8x³.

Solution. We ask ourselves the question: What function has as its derivative 8x³? To answer the question we think in terms of our memorized differentiation rules and formulas and work backward, trying to figure out what function, when differentiated, would give 8x³. The answer is 2x⁴ + c where c is any constant.

Definite integral. The definite integral is the most basic, fundamental concept of integral calculus.

Consider the function y = f(x) shown in Fig. 1. We wish to find the area A bounded by the curve y = f(x), the x-axis and the lines x = a and x = b. To find this area we proceed as follows: We divide the interval [a, b] into n sub-intervals, not necessarily of equal length. We denote the length of the first subinterval by Δx₁, of the second by Δx₂, and so forth up to the final subinterval Δx_n. In each subinterval Δx_i we arbitrarily choose a point x_i and set up the sum

1) A_n = f(x₁)Δx₁ + f(x₂)Δx₂ + ... + f(x_n)Δx_n .

A_n is thus equal to the sum of the areas of the rectangles shown in Fig. 1 and represents an approximation to the area A that we wish to compute. We now define the area A to be the following sum:

in which we require that the largest of the sub-intervals, max Δx_i, approach zero as the number of sub-intervals n → ∞. We thus make an infinitely fine net on the interval [a, b] in which the largest sub-interval is required to approach zero.

We have assumed that f(x) 0. If f(x) changes sign, then the limit 2) will give us the algebraic sum of the areas of the segments lying between the curve y = f(x) and the x-axis, where the segments above the x-axis are taken with a plus sign and those below with a minus sign. See Fig. 2.

The need to calculate the limit 2) arises in many problems. For example, suppose that a point is moving along a straight line with variable speed s = s(t). How do we determine the distance d covered by the point in the time from t = a to t = b?

Let us assume that the function s(t) is continuous. Let us divide the interval [a, b] on the t axis up into n sub-intervals, of length Δt₁, Δt₂,... , Δt_n. To calculate the approximate value for the distance covered in each subinterval Δt_i, let us suppose that the speed during the period of time is constant i.e. it is equal throughout the subinterval to its actual value at some intermediate point t_i. The whole distance covered will then be expressed approximately by the sum

and the exact value of the distance d covered in the time from a to b, will be the limit of such sums for finer and finer subdivisions; that is, it will be the limit

The example just given is only one of many that could be given. Many practical problems lead to the calculation of such a limit.

The limit 2) above is called the definite integral of the function f(x) taken over the interval [a, b].

Note. There is a natural question that a reader might ask. Why such a complicated definition? Why not just define the sub-intervals to be equal with the x_i’s placed at the midpoints? I can’t give an answer to that. The definition we have given appears to be the standard, universally used definition.

Def. Definite (Riemann) integral. The definite integral of a function f(x) taken over the interval [a, b] is denoted by

and is defined as

where f(x)dx is called the integrand and a and b are the limits of integration; a is the lower limit, b is the upper limit.

Note the source of the notation for an integral by comparing the left and right members of 3) above: dx corresponds to Δx, f(x) corresponds to f(x_i), and the integral sign ∫ represents the letter S, meaning “sum”.

The intuitive interpretation of a definite integral

is that of the area bounded by the function f(x), the x-axis and the lines x = a and a = b.

The connection between differential and integral calculus. There is a profound connection that exists between differential and integral calculus, a connection which provides us with a general method of calculating definite integrals for an extremely wide class of functions.

To explain this connection let us give the following example from mechanics: Let us suppose that a point is moving along a straight line with a speed s = s(t), where t is time. We have already shown that the distance Δd covered by our point in the time between t = t₁ and t = t₂ is given by the definite integral

Now let us assume that we know the function d = d(t) giving the distance traveled as a function of time as calculated from some initial point A on the straight line. The distance covered in the interval of time [t₁, t₂] is then given by

5) Δd = d(t₂) - d(t₁)

Thus, from 4) and 5), we have

Now the function d(t) is a primitive of s(t) i.e.

Thus 6) provides us with the connection between differential and integral calculus. The relationship 6) tells us that we can compute the definite integral

by finding its primitive d(t), its value being given by d(t₂) - d(t₁).

From this example from mechanics we are led to the well-known Fundamental Theorem of Integral Calculus:

Fundamental Theorem of Integral Calculus. If f(x) is continuous and F(x) is any arbitrary primitive for f(x) i.e. any function such that F'(x) = f(x), then

This is often written as

where

is a shorthand notation for

F(b) - F(a).

Thus, by this theorem, the problem of calculating the definite integral of a function is reduced to the problem of finding a primitive of the function.

Credit for The Fundamental Theorem of Integral Calculus has been given to Isaac Barrow, the principal teacher of Sir Isaac Newton.

Area function, A(x). The definite integral

represents the area under the curve f(x) between lines x = a and x = b where a and b are regarded as fixed numbers. The integral represents some number, namely the measure of that area. Now let us consider the upper limit b of the integral to be variable. Let us replace b with x. The integral then becomes

and represents a variable area A(x) extending from x = a to a variable point x. In other words,

where A(x) is the area shown in Fig. 3. This area function A(x) is a primitive of the function f(x) i.e. dA/dx = f(x). That this is true is quite obvious. It is quite easy to see that

dA = f(x)dx

and consequently that dA/dx = f(x). A small increase in x of Δx will produce an increase ΔA in

area of approximately f(x)Δx. See Fig. 4. Thus

ΔA f(x)Δx

which we can rewrite as

and

Now let F(x) represent some particular primitive in the family of primitives of f(x). Then, since A(x) is also a primitive,

10) A(x) = F(x) + c

where c is a constant. We can evaluate c by substituting in known values: the value of A(x) corresponding to the initial point x = a is zero. Hence, substituting into 10),

0 = F(a) + c

c = - F(a) .

Replacing c by - F(a) in 10) we have the expression for the area from x = a to any other point x :

11) A(x) = F(x) - F(a)

Finally, the area under the curve from x = a to x = b is found by replacing x by b so

A = F(b) - F(a) .

We see that 12) is the same equation as 7) above and we have derived the Fundamental Theorem of Integral Calculus.

The area function A(x) of some function f(x) is conceived of as the measure of an area generated by an ordinate of variable length which starts at x = a and moves to the right, its upper end always on the curve. The area generated when the moving ordinate has reached any point x is a quantity which depends on x; i.e. it is a function of x.

Problem. Given an arbitrary function f(x), one not necessarily expressed in terms of formulas or analytically, perhaps one in tabular or graphical form. Does a primitive for it exist? If so, exhibit it.

Solution. The area function A(x) representing the area under the curve extending from an arbitrary starting point “a” up to the value x constitutes a primitive for it. It can indeed be proved that if a function f(x) is continuous (and even if it is discontinuous, but Lebesgue-summable) a primitive F(x) satisfying 7) above exists.

Properties of definite integrals. If f(x) and g(x) are continuous on the interval of integration a x b the following hold:

6. The First Mean Value Theorem: