§ 20. SUCCESSIVE DIFFERENTIATION.

At § 17 it was explained how it was sometimes necessary to differentiate a derived curve. The primary curve in that case was a time-distance curve, the first derived a time-velocity, and the second derived a time-acceleration curve. Now, the height of the primary being denoted by y, and that of the first

dy

derived by y or the height of the second derived may,

dx

on the same principle, be denoted by y" or

however, is very inconvenient to write and print. It is therefore shortened by treating it as a simple fraction in which d stands for some definite algebraical quantity. (In reality, of course, it does not mean anything of the kind.)

Thus we

The student must be careful to notice that this quasi

expression has absolutely nothing to do with x, and that at present he may regard das merely symbolical. In the same way, the height of the third derived is—

§ 21. NOTATION OF INTEGRATION.

It will be seen from § 13 that the process of graphical integration consists of a construction whereby a curve is obtained of which the tangent of angle of slope is at all points

ordinate of curve we wish to integrate. The algebraical process is the exact counterpart of this, and consists in obtaining an expression which, when differentiated algebraically, will give as a result the expression which we wish to integrate. There is no general method of performing this reverse operation. Indeed, in a great number of cases it cannot be performed at all except by the aid of an infinite series. We are in all cases obliged to rely on our previous experience of differentiation. If the expression is of a type of which we have had no previous experience, we cannot do anything with it until we have twisted it into a shape which we do recognize as the result of some differentiation with which we are already acquainted.

Suppose, for instance, we wish to integrate 3x2. This means that y 3x2 is to be the first derived of the curve we wish to find. The problem is stated thus for an "indefinite

=

These expressions will be presently explained.

This symbol may be regarded in two ways: (1) It may be taken simply as a question mark. The meaning then is

S

3x2

d

x

? expression will give 3x2 when differentiated with respect to x (2) It may be taken to be the letter s, the first letter of the word "sum," thus

b

3x2dx

α

The sum between the ordinates (x=6) and (x=a) of all such rectangles as 3x2dx For it is evident that the area in square inches of a very thin vertical strip of the curve such as that shown in Fig. 21. is 3x2dx (see §§ 13 and 17), and the sum of all the thin strips into which the area between 6 and be divided a may = whole area of curve between these two ordinates = difference between corresponding ordinates of upper curve, as already explained (§ 13).

x

3.2.2 ↓

X

FIG. 21.

Now, let us consider what expression will give 3x2 when differentiating with respect to independent variable. (The student will understand the last expression better after reading the next chapter.)

Consider what is the rule just proved (§ 17) for differentiating ". We have found the differential coefficient to be nxn−1).

Hence the answer to the question fnx-dx is a". It is easily seen that the given expression 32 is of the form nam-1) where n = 3, hence—

f 3x2dx = x3

A more complete solution, as will be presently explained, is "+some constant."

Hence we see that 3x dx = a is exactly the same

3x2, but put into another form. In just

This is sometimes symbolically expressed by saying that f and a "cancel one another." Thus multiplying both sides

of the equation

d(x)

dx

d(x3) = 3x2dx

Now multiply by. We obtain

fd(x) = f 3x2dx

or, since and d cancel

x = √ 3x2dx

This is not altogether a happy analogy, for and d do not cancel on the right-hand side. The idea is that if any quantity, A (represented here by the ordinate of the upper curve), be divided into a large number of parts, and then all the parts be added together, the quantity A is reproduced.

22. THE "CONSTANT" IN INTEGRATION.

The expression "x3+constant" is known as the "indefinite integral" of 3x2. It is a general expression for the height of every possible primary which has y = 3x2 for its first derived. We have already seen (§§ 8, 13, etc.) that there are an infinite number of such curves corresponding to different starting-points on the line OY. If a value K (suppose) be assigned to the "constant," the value of + K at any point also represents absolutely the area of the curve y' that ordinate and the ordinate corresponding where the curve y = x2 + K cuts the axis of x. may be found by putting y = o in the equation and solving for x. Thus here

x = - K

=

3x2 between to the point. This point

This may be generally explained as follows. Suppose we have any curve P'Q' (Fig. 22) of which the equation is y' =ƒ'(x) (where f'(x) is a shorthand symbol for “any expression containing "), and suppose, having integrated it, we obtain a curve PQ or TK, or some other parallel curve of which

the equation is y = f(x)+c, where f(x) is some different expression containing x. Then, in accordance with the notation already explained, we have ff(x)dx = f(x) + constant. Then the expression f(x)+c is called the indefinite integral

=

of f(x). Let x = Oq. Then for some particular value of c, f(x) + c 2Q. For some other value of c, f(x) + c = qK. Since the slope at K is the same as the slope at Q, the height q'Q' is the same in each case. Now, qQ = area of lower curve between the ordinate p'P' (corresponding to the point