The integral calculus hats thus performed for Its a wonderful task. speaking, it employed aeylinder v[ditine r tilled with gas and compressed it, an infinitesimal amount at at time. until unit volume attained: it then summed up the infinitesimal amount, of wo1: a 1111 it has 10111 that the 11.1:11 amount of work elouo is •. The result is algebraic. sine,. I. may hale any finite value whatever. But to obtain at desired particular result. all we have to do is to substitute for a some numerical value. Further. the work per formed in compressing the gas from par tieular volume I., to some particular volume r, e% Wilily equals the differ, arc between the work required to reduce the gas from volume r, to unit volume, and that required to reduce the gas from volume r. to tiiiit volume. The required work, \V, between the limits I.:, and is, there fore, W = c — — c.
The limits being defined, this 'integral' is called a Definite Integral. and the operation is usually denoted as follows: e, = The constant k, coining from the law pr = k, depends on the amount of gas employed and on the temperature at which the compression is carried out. It may, of course. be found by at-wally measuring the pressure and volume of the given amount of gas at the given tempera ture, and multiplying the pressure by the vol ume. By substituting in the above expression this value of k, as well as the numerical values i the initial volume r, and the final volume involved in an actual compression, we will finally obtain the work which the problem required to calculate, and this is actually the way in which engineers determine that important quantity.
.4 ?Tither Way of Stating and Solving the Third Problem.—The relation pr = k between the pres sures and volumes of a gas whose temperature is kept constant (i.e. the law of Boyle and \lari otte) may he represented geometric-ally by a curve called an equilateral hyperbola, every point of the curve corresponding to a definite pressure and volume. (See ASYMPTOTE.) Fur ther, it is shown in text-books of natural philoso phy that the work performed in compressing the gas from an initial volume represented by the line (1.1 (see figure) to a final volume, OB, is represented by the area A.1'1313. The problem of determining the work may therefore be viewed as requiring to determine the area inelosed by the hyperbola and the axis OV between the limits ()A = r, and OB = this problem the area AXIVII may be imagined as made up of an infinite number of infinitely narrow strips. (inc
such strip is roughly shown in the figure between the lines marked p and p'. The difference he tween p and p' would evidently be the greater, the greater the distance between them. lint since thy is supposed to be infinitely small, the two lines may be taken as equal and the -trip may Is' considered a. a reetangle. falling its infinitely small base dr, the area of the reetangle is seen to be pdr. The total area AA'Ini may now be obtained by summing up the infinite 1111 Oi lier of 'differentials of area' like pdr inclosed between the limits OA = r, and OB = The summation may be performed by the integral calculus and is denoted by a definite integral, as follows: AA'B'B = f pdr.
r., We have seen before that the result of this integration is The required area there . 2 fore equals the natural logarithm of the ratio OA —, multiplied by the number (k) representing OB the product of any pair of coiirdinates, such as 011 X lig', or OA X AA'. etc. The calculus method is analogously employed whenever it is required to find the area inclosed by a given curve, elementary geometry being in most cases powerless to furnish the desired answer; and thus the calculus finds exten.:ive application iu the solution of many important problems of geometry.
The above sketch outlines the methods of rea soning by which the calculus attacks problems involving variable quantities. As to its limita tions, it must be observed that while the differ ential calculus teaches how to differentiate read ily any function whatever, the converse problem, viz. that of integrating a given differential, is often very difficult, requiring all manner of alge braic artifiees, and is sometimes altogether im possible. In other words, in their studies of nature, scientists are often led to construct dif ferentials ( just as in our third problem we con structed the differential pdr) which they cannot integrate, because they can conceive no function which, on differentiation, would yield the given differential. Finally, it may be observed that the flour is no better than the grain, and if data that are made to pass through the mill of the calculus lead to doubtful results, it is the fault not of the calculus, but of the data : the calculus itself is as exact as any other branch of mathe matics, in spite of the fact that the things it deals with seem so often to dwindle away into nothing.