area and thickness ar, three long prisms of length x and section (A)2, and one small cube of volume (▲). From the above, we have

which in the limit (ie. when Ar becomes infinitely small) be

By a similar process, we can show that if s = 1a —

Likewise, if we expand by the binomial theorem, or if we work out a lot of results and tabulate them, we shall find that the following relation holds :

where A and C are constant quantities, and which of course do not vary; then, if x increases by an amount ▲r, and in consequence y increases by a corresponding amount ▲y, we have

x + Ax = A(y + ▲y)2 + C

=A[+2yay + (Ay)] + C

Subtracting the original value of x, we have

It should be noticed that the constant C has disappeared, while the constant A is multiplied by the differential of y2. Similarly for the constants in the following case.

Letr Apm± Byn Cy± D, where the capital letters are

constants. Then by a similar process to the one just given, we get

It sometimes happens that x increases to a certain value, its maximum, and then decreases to a certain value, its minimum, and possibly increases again, and so on. If at any one instant it is found to be increasing, and the next to be decreasing, it is certain that there must be a point between the two when it neither increases nor decreases; this may occur at either the maximum or the

dx dy

minimum. At that instant we know that = o. Then, in order

to find when a given quantity has a maximum or a minimum value,

dx
dy

we have to find the value of and equate it to zero.

Thus, suppose we want to divide a given number N into two parts such that the product is a maximum. Let be one part, and N- - r the other, and y be the product. Then

N

Thus we see that y has its maximum value when a =

2

= 5.

In some cases we may be in doubt as to whether the value we arrive at is a maximum or a minimum; in such cases the beginner had better assume one or two numerical values near the maximum or minimum, and see whether they increase or decrease, or what is very often a convenient method-to plot a diagram. This question is very clearly treated in either of the books mentioned above.

The process of integration follows quite readily from that of differentiation.

Let the line ae be formed by placing end to end a number of

41

short lines ab all of the same length. When the line gets to c, let ts length be termed L, and when it gets to e, L. The length ce we have already said is equal to ab. Now, what is ce? It is simply the difference between the length of the line ae and the line ac, or L.-L.; then, instead of writing in full that ce is the difference in length of the two lines, we will write it Al, i.e. the difference in the length after adding or subtracting one of the short lengths. Now, however long or however short the line may be, the difference in the length after adding or subtracting one of the lengths will be Al. The whole length of the line L. is the sum of all the short lengths of which it is composed; this we usually write briefly thus:

FIG. 616.

or, the sum of (x) all the short lengths Al between the limits when the line is of length L. and of zero length is equal to Le. Sometimes the sign of summation is written

Similarly, the length L is the sum of all the short lengths between b and c, and may be written

the upper limit Le being termed the superior, and the lower limit L the inferior limit: the superior limit is always the larger quantity. The lower limit of is subtracted from the upper limit.

In the case of a line, the above statements are perfectly true, however large or however small the short lengths Al are taken, but in some cases which we shall shortly consider it will be seen that if Al be taken large, an error will be introduced, and that the error becomes smaller as ▲l becomes smaller, and it disappears when al becomes infinitely small; then we substitute dl for Al, but it still means the difference in length between the two lines Le and Le, although ce has become infinitely small. We shall now use a slightly different sign of summation, or, as we shall term it, integration. For the Greek letter Σ (sigma) we shall use the old English s, viz. s, and—

The expression still means that the sum of all the infinitely short lengths di between the limits of length L. and L is L1, which is, of course, perfectly evident from the diagram.

Now that we have explained the meaning of the symbols, we will show a general connection between the processes of differentiation and integration. We have shown that when

But the sum of all such quantities dr, viz. fdx, = x = y" (iii.) ; hence, substituting the value of dx from (ii.), we have—

This operation of integrating a function of a variable y may be expressed in words thus:

Add to the index of the power of y, and divide by the index so increased, and by the differential of the variable.

In order to illustrate this method of finding the sum of a large number of small quantities we will consider one or two simple cases.

Let the triangle be formed of a number of strips all of equal length, ab or Al. The width w of each strip varies; not only is each one f different width from the next,

FIG. 617.

but its own width is not

constant. If we take the greater width of, say, the strip cefi, viz. We, the area w▲l is too great, and if we take the smaller width we, the area weal will be too small. The narrower we take

the strips the nearer will wewe, and when the strips are infinitely narrow, W2 = We = W0, and the area will be wodl exactly, and the stepped figure gradually becomes a triangle.

The area of the triangle is the sum of the areas of all the small strips w. dl (Fig. 618). Hence the area

But

W
W L'

or w =

WI
L'

of the triangle is the sum of all the small strips

dl, which we write

FIG. 618.

WI

of area

L

W

The is placed outside the sign of integration, because neither

L

W nor L varies. Now, from other sources we know that the area

of the triangle is

WL
2

Thus we have an independent proof of the

accuracy of our reasoning.

Suppose we wanted the

area of the trapezium bounded by

W and W1, distant L and L, from the apex. We have

But here again we know, from other sources, that the area of the trapezium is—

which again corroborates our rule for integration.

By way of further illustrating the method, we will find the volume of a triangular plate of uniform thickness t. The volume of the plate is the sum of the volumes of the thin slices of thickness dl.