CHAPTER VI. THE APPLICATION OF SINGLE INTEGRATION TO QUESTIONS OF GEOMETRY. SECTION 1.-Rectification of Plane Curves referred to Rect angular Coordinates. 154.] In the present Chapter I propose to consider some of the most simple applications of single integration to questions of geometry, reserving the more complex problems of space until the higher parts of the Integral Calculus, which are required for their solution, have been more fully developed. And, in the first place, I will shew how this calculus enables us to determine, either exactly or approximately, the length of a plane curve between two given points in terms of the coordinates of these points. Its length will thus be compared with that of a straight line; and hence the process has been called the Rectification of Curves. Let y = f(x), or F(x, y) = c, be the equation of a plane curve referred to rectangular coordinates; and let it be required to determine the length of the curve between the points (x, yo) and (Xn, yn); that is, to determine the length of a straight line, along which, if the curve is made to roll (not to slide), the extremities of it will coincide with those of the curve. Now, adopting the notation of Vol. 1, Art. 218, let ds be an infinitesimal length-element of the curve, then the required length is the integral of ds between the specified limits; but ds = {dx2 + dy?}t; and therefore the required length = /{dx+ + dy?}); (2) the integral being taken between the given limits. Let s represent the length of the curve; then if the equation is y = f(r), so that dy = f'(x) dx, de = {1+(f'(x)){}« dx ; (3) (1) And if the equation to the curve is of the form x = f(y), so that dx = f'(y)dy, then 8 = 1 " {1+(f"(y))2}}dy. (4) As the radical expression in (1) involves an ambiguity of sign which is continued in (3) and (4), and as s is an absolute length, we must choose that sign which the circumstances of the problem require; that is, ds and dx or dy must be taken with the same or different signs according as x and y increase or decrease when s increases. Although I have given the general formulæ (3) and (4) which express the length of a curve as a definite integral, yet in the sequel, as the following examples shew, it will be more convenient to deduce the expression for s directly from the equation to the curve. 155.] Examples of rectification of plane curves. Ex. 1. The circle; see fig. 3. Let the centre be the origin; and let the arc APM, whose length is required, begin at a, and be measured from a towards B; so that, if ap=s, OM= x, x decreases as s increases ; let om, = xn; then since x2 + y2 = a”, ds =-ade (a? — x2) x, Ja da ... the length of the arc 2-22) Mng example is a definite into) and (4) wh; a. The perimeter of the circle = 27a. Hence also if om, = Xo, then fxm - adx the arc P.Po = 1. consequently the whole length of the cycloid is 8a; that is, four times the diameter of the generating circle. Since (11) expresses a general relation between the length of an arc of the cycloid measured from the vertex and the abscissa to the extremity of that arc, it may be and frequently will be employed hereafter as the equation to the cycloid. If oqa, fig. 5, is the generating circle of the cycloid, the equation shews that the arc OP = twice the chord OQ. (2) Let the starting point be origin; see fig. 6, and let OM = x, MP = y; OM, = Xn M, P, = Yn; so that = 4a; which result is the same as that found in the former case. Ex. 4. The Tractory; see fig. 2. Let the required arc begin at A; and let the ordinate at its extremity be Yn; then since -dy dx ds. the required arc = /* -gang = [-a logy) = a logo and writing for y, the general value y, we have 8 = alog, (14) (15) |