Pocketbook of Useful Formulæ and Memoranda
for Civil and Mechanical Engineers

625 OF ENGINEERING FORMULÆ. Differential and Integral Calculus. Paragraphs.which refer to the Differential Calculus are in italics; those which refer to the Integral Calculus are in full-face type ; those which refer to both are in ordinary type. The object of the Differential Calculus is to find how the indefinitely small changes in some variable quantity alter at each instant the value of a quantity dependent upon it. The object of the Integral Calculus (the reverse of the Differential) is to ascertain from the ratio of indefinitely small changes in. two or more mag- nitudes the function (f) which, governs the changes. Notation. Constant quantities, which retain the same value throughout the investigation, are usually represented by the early letters of the alphabet—a, b, c, e, &c. Variables, to which different values may be assigned, by the later letters, u, v, w, x, y, z. The latter are frequently (but not invariably) used to denote the following : w = one or more functions; sometimes w = length ; v = volume; x = abscissa; y — ordinate; z = surface, or area ; d = differential, or the sign of differentiating; f is the sign of integration of the quantity that follows it: fff— successive integration; J"“ denotes that the integration is to be within the limits of a and b. Rules for Differentiation and Integration Rule for Differentiation of any power of the variable x.—Deduct 1 from the index of the variable, and multiply by the original index, or d x>‘ = n xn~l.dx. For example, dax'A — 3 a æ3-1 — 3ax^ ,dx. Kule for Integration.—Acid 1 to the index of the variable, and. divide by the new inclex,*or/x"dx x11 +1 3 a x* + 1 „ = —--T • For example, f 3 a x2 d. x - ■ -------------= a x3. n t 1 o A constant, if a coefficient to a var iable, is unchanged in differen- tiating ; thus d. a xS = 3 ax'S. dx. If the constan tbea term, it disappears; thus d(a + x'sj — 3 x%. d x. A constant factor may be removed from the process of integration, thus, / a d x = adx. A constant term must reappear in integration in the form of an arbitrary constant, thus, f 3 x2 d x = x:i + C. f and d neutralize each other, thus,/, dx = x. * Except when n = — 1; then j x - 1 dx = J" — = log.^x. 2 S