In fig. 23 the sum of the two areas rafined over thus and ||||| is evidently (X2-X11). The first of these areas, namely, that between the curve, the vertical axis, and the two horizontal lines at levels X1 and X2, is S xdX. The second, included X1 between the curve, the horizontal axis, and the two verticals at X1 Xdx. Hence the above equation. In fig. 23 the point 3 has a negative ordinate x, while its Xordinate is positive. A useful exercise for the student is to follow the variations of the proposition, taking the pair of points in each possible pair of quadrants of the full diagram, 1 CHAPTER V. PARTICULAR LAWS. 102. Any Power of the Variable.-In § 99 the integral of was found to be log, x. This is the single exception to a general law giving the integral of any power of a variable, special examples of which have already been demonstrated in §§ 59, 61, 71, 79, and 86, namely, the integrals of the 0th, 1st, 2nd, (-2)nd, 5th, (-4)th, 8th, and 31th powers. All these examples conform to the general law 103. Any Power of the Variable by Logarithms. This result can be proved to be always true by logarithmic differentia * See Classified List, III. A. 2. tion as explained in § 100; thus, n being any power, positive or negative, integer or fractional, 104. Diagram showing Integral of x-1 to be no Real Exception. This formula is true for all powers of the variable, with the one exception of the integral of or x-1, which is log, x. 1 х 1 It is often a puzzling question to students why there should be this one solitary exception to so general a law. Fig. 24 has been drawn to demonstrate graphically that it is only formally an exception; that it is in reality no exception at all. To compare this one apparent exception with other cases of the general law, the integrations must be taken between the same limits. It may be convenient to take the lower limit of x equal to 1 because log 1=0 and [log æ]=log. Taking the same lower limit for 1 1 because 1" 1 whatever n be. xn-1 In fig. 24 there are drawn to the scales shown, the curves n for n=2, 1,,, -, -, -1, and -2. There is also plotted to the same scales the curve log, x. It will be seen that the curves for n= and n = lie very close together, and that the curve log, a lies between them throughout its whole length. This shows that the logarithmic curve is simply one of the general set of curves illustrating the general law, and that it is no real exception to the general law. Its position between the curves for n= ±10 shows that log, x is simply the special name given to the value of xn-1 the function when n is an excessively minute fraction, or rather n when n is zero. Considering the variation of the curve in fig. 24 downwards from positive values of n to negative values of n, it is clear that the curve must have some definite position as n passes through zero, a position lying between that for small positive values of n and small negative values of n. This position is that by a special method, the result appearing in a special form. It should be noted that all the curves pass through the height 0 at the horizontal distance x=1, and that they have here one common tangent or gradient = 1. 105. Any Power of Linear Function. If a, b, and n are constants, and X = (a + bx)", = n(a + bx)n-1 × b=bn(a + bx)~~1. (a + bx)n+1+C* Written inversely for integration, this is 1 (a + bx)ˆdx : = b(n + 1) the constant C being introduced by the integration. 106. Reciprocal of any Power of Linear Function. This last integration rule fails when n= – 1. In this case we find by §§ 51 and 98, [(a + bx) 1 bx) ̄1dx -1dx ==== b loge (a+bx)+C= 2.3--b 2·3- ̄ ̄ log10 (a + bx) + C .† ах 107. Ratio of Two Linear Functions.-The function be reduced so as to make it depend on the last case, 108. Ratio of Two Linear Functions; general case. Since the function A+ Bx A Bx = * See Classified List, III. A. 4. See Classified List, III. A. 5. the integration of this function is performed by combining th results of §§ 106 and 107. 109. Quotient of Linear by Quadratic Function. X=a+bx2; then X'=2bx and = 1 X' a+bx2 2b X'. Now d therefore loge (a + bx2) + C I X XC dx= where C is the integration constant. Similarly if X=a+bx + cx2; then X'=b+2cx, and, therefore, any function of the form A+ Bx can be readily integrated by splitting it into two terms as in § 107.* 110. Indicator Diagrams. An important case of the use of the law of S$ 102 and 105 is the integration of the work measured by an indicator diagram. If at any stage of the expansion p be the pressure and v be the volume of the working substance, then as the volume increases by dv, the work done is pdv. Taking the expansion law in the more general form of § 105, or p=(a+bv)"; then the work done during expansion from P1, v to a v V1 P2, V2 dv is b(1 n) [p(a + bv) ]; [ Here the index is always negative. If it is arithmetically greater than 1, the expansion curve makes p(a+b) negative. 1 But at the same time the divisor (1-n) is negative, so that the formula makes the work done positive. It is then better to reverse the limits and to use the positive divisor (n − 1). * See Classified List, III. A. 23. |