angle may be expressed in terms of its sine as in last article. If s be the sine, we have Sectorial area = r2 sin1s. In fig. 13 this is the area NaO; in which figure the length as measures rs of the present article, and acr cos a=r√1-2 of the present article. The triangular area aOc, therefore, equals r2s/1-2. Add this to the above sectorial area; the sum is the area ONac. This area may be taken as made up of a large number of narrow strips parallel to ON, the height of each of which would be cos a=r√1-82, while the horizontal width would be r.ds. The area OÑac is, therefore, the integral of this narrow strip of area from ao to a=a, which limits correspond to from so to s=s. Twice this is the area of a circular zone lying between a diameter and a parallel at the height s from the diameter, the radius being assumed 1 in the last equation. Here, again, & cannot range outside the limits ±1. 114. x(r2x2)- integrated. The function x(x2-x2)- may be looked on as the sine divided by the cosine, i.e., the tangent of an angle, see fig. 13, while dx is the increment of the sine. The increment of the sine multiplied by the tangent evidently equals the decrement of the cosine, and accordingly the integral is minus the cosine, or (p2 − x2)1.† 115. (22) integrated. The function (2+2)- is more difficult to deal with. Let X represent any function of x, and multiply and divide its reciprocal by (x+X); thus :— or integrating x2+k=X2 or X = (x2 + k)1 where the integration constant k may be either + or kr2, we have Here, if k is negative, the differential is "imaginary" and cannot occur in any physical problem except for values of a greater thank. 116. {x(2x2)} integrated. The integral of x(1·2 — x2) ̄‡ is found most easily by substituting for x. Thus 1 1 X x-1; x - - - - - x2; - - - dx. X 2 dx X2; 117. Log integrated. The integral of the logarithm of a variable number N is found by help of the formula of reduction in § 80.and by § 98, thus: loge is the "modulus" of the system of logarithms whose base is b, and for the decimal system is 0.4343 nearly. Therefore, log1o NdN = N{log1, N-4343} + C.‡ * See Classified List, III. B. 6, 3, and 4. 118. Moment and Centre of Area of Circular Zone.With the notation already used, we saw in § 113 that a narrow strip of the area of a semicircle is 2r2(182)ds. The distance of this strip from the diameter from which the angle and its sine are measured is rs, and the product of the area by this distance is the moment of the strip-area round this diameter. This is 27-38(1-82)ids, in which is a constant while s varies. Since s = d(1-82) the integral moment of all the strips for a zone between the diameter and a parallel rs away from it is easy to find. Calling (1-82) by letter S, we find Integral moment = 2r3 (s(1 − s2)ids when taken from lower limit s= 0 or S = 1. From this is deduced by dividing by the area of the zone; For the whole half circle, sin-1's becomes a right angle or 22 2 while s= 1 and (1-2)=0. Therefore the centre of area of a semicircle is distant from the centre of the circle by The moment of the whole semicircular area round the diameter is 3. The integration performed here is a geometrical illustration or proof of the general integral of x(1 − x2)* 119. (2+2)-1 integrated. In § 78 it was found that the angle-gradient of the tangent equals the square of the reciprocal of the cosine. Remembering that Here t is essentially a number or pure ratio, and it may vary from - ∞ to +∞o. If, in order to make the formula of more general application, we introduce a constant 2 as follows, then t may be any+or physical quantity, but t and r must be of the same kind. Then, since (૩) કે dt r dt 1 r2+t2 2° tan+C.t 120. (2-2)-1 integrated.-Since (2-2)=(r+x) (r-x), we find easily 1 1 1 1 and therefore this function can be integrated by help of § 106. 121. Hyperbolic Functions and their Integrals. The functions (ee) and (e-e-*) enter largely into the geometry of the hyperbola and of the catenary, as well as into the investigation of several important stress and kinetic problems. From the origin of their utilisation by mathematicians they are called hyperbolic functions, and from certain useful analogies between the geometry of the hyperbola and that of the circle, the names sinh x and cosh x are applied to them.‡ In § 96 we have already found the x-gradient of e. Using it, we obtain Scosh sinh x dx = cosh x . The integrals of other hyperbolic functions are equally easy to find, and are tabulated in Section V. of the Classified Reference List at the end of this book. ! CHAPTER VI. TRANSFORMATIONS AND REDUCTIONS. 122. Change of Derivative Variable. From the fundamental idea of X', the x-gradient of the function X, as explained in § 35, we have in equations between increments or between integrals X dX リ Now, it may be easier to find X than to find Xda, and it amounts to the same thing. The former integration may be easier if X' be capable of convenient expression in terms of X. This transformation has been frequently employed already, as, for instance, in §§ 119, 118, 116, etc., etc. More generally, if f(X) be any function of X when X itself is a function of x, it may facilitate the integration to substitute provided either that X' cancels out of the expression that X' is expressible in terms of X. 123. Substitution to clear of Roots.—A selection of such transformations is given in Section II. G of the Classified List, under the title of Substitutions. |