Now diminish indefinitely each term of which h is the type, then Ehp vanishes, and we have finally that is [{(x, y) dy} dr. This is more concisely written dy being placed to the right of dox because the integration is performed first with respect to y. 61. We may again remind the student that writers are not all agreed as to the notation for double integrals. Thus we use [*[* (x, y) dx dy to imply the following order of oper a α ations-integrate (x, y) with respect to y between the limits a and B; then integrate the result with respect to x between the limits a and b. Some writers would denote the same order of operations by [*[* (x, y) dy dx. a α 62. We might have found the limit of the sum in Art. 60 by first taking all the terms in one vertical column, and then taking all the columns. In this way we should obtain as the 63. Hitherto we have integrated both with respect to x and y between constant limits; the limits however in the first integration may be functions of the other variable. Thus, for example, the symbol (x, y) dx dy will denote the following operations-first integrate with respect to y so that x a x(x) is constant; suppose F(x, y) to be the integral; then by taking the integral between the assigned limits we have the result F{x, f(x)} – F{x, x (x)}. We have finally to obtain the integral indicated by [ ̊ [F {x, y (x)} – F{x, x(x)}] dx. a The only difference which is required in the summatory process of Art. 60 is, that the quantities a, YYYm-z will not have the same meaning in each horizontal line. In the (r+1)th line, for example, that is in 3 hr+1 {k1Þ (x,, α) + k2 † (xr, y1) + k ̧¤ (Xr, Y2).........+ km & (Xr, Ym_1)} we must consider a as standing for x (x,), and y1, y2...... as a series of quantities, such that x(x), Y1 YYm-1 ¥ (xr), are in order of magnitude, and that the difference between any consecutive two ultimately vanishes. Hence, proceeding as before, we get (x, y) dy for the limit of the sum of the terms in the (+1)th line. (4(xr) x (xr) 64. It is not necessary to suppose the same number of terms in all the horizontal rows; for m is ultimately made indefinitely great, so that we obtain the same expression for the limit of the (r+1)th line whatever may be the number of terms with which we start. 65. When the limits in the first integration are functions of the other variable we cannot perform the integrations in a different order, as in Art. 62, without special investigation to determine what the limits will then be. This question will be considered in a subsequent chapter. 66. From the definition of double integration, it follows that when the limits of both integrations are constant, [[$ (x) 4 (y) dx dy = [$ (x) dx × f† (y) dy, supposing that the limits in (y) dy are the same as in the integration with respect to y in the left hand member, and the limits in (a) do the same as in the integration with respect to x in the left hand member. For the left hand member is the limit of the sum of a series of terms, such as and the right hand member is the limit of the product of h ̧‡ (x) +h ̧¢ (x ̧) +h ̧‡ (x2)................+ h2 $ (Xn-1), 1 2 and k ̧¥ (y) + k2 † (y1) +k ̧¥(y2)................+ km ¥ (Ym-1)• 67. The reader will now be able to extend the processes given in this chapter to triple integrals and to multiple integrals generally. The symbol will indicate that the following series of operations must be performed-integrate (x, y, z) with respect to z between the limits and; next integrate the result with respect to y between the limits 7, and 7,; lastly integrate this result with respect to x between the limits & and 1. Here and may be functions of both x and y; and 7, and 7, may be functions of x. This triple integral is the limit of a certain series which may be denoted by Zo (x, y, z) Ax ▲y Az. no 71 CHAPTER VI. LENGTHS OF CURVES. Plane Curves. Rectangular co-ordinates. 68. Let P be any point on the curve APQ, and let x, y be its co-ordinates; let s denote the length of the arc AP measured from a fixed point A up to P; From the equation to the curve we may express terms of x, and thus by integration s becomes known. 69. The process of finding the length of a curve is called the rectification of the curve, because we may suppose the question to be this: Find a right line equal in length to any assigned portion of the curve. In the preceding article we have shewn that the length of an arc of a curve will be known if a certain integral can be obtained. It may happen in many cases that this integral cannot be obtained. Whenever the length of an arc of a curve can be expressed in terms of one or both of the coordinates of the variable extremity of the arc, the curve is said to be rectifiable. 70. Application to the Parabola. The equation to the parabola is y = √(4ax); hence = √/ (ax + x2) + a log {√x+ √√(a + x)} + C. = Here C denotes some constant quantity, that is, some quantity which does not depend upon x; its value will depend upon the position of the fixed point from which the arc s is measured. If we measure from the vertex then s vanishes with x; hence to determine C we have a log √a + C = 0; and thus s = √(ax + x2) + a log {√/x+√ (a + x)} − a log√a If then we require the length of the curve measured from the vertex to the point which has any assigned abscissa, we have only to put that assigned abscissa for x in the last expression. Thus, for example, for an extremity of the latus rectum x= = a; hence the length of the arc between the vertex and one extremity of the latus rectum is a√2+ a log (1+√√/2). 71. In the preceding article we have found the value of the constant C, but in applying the formula to ascertain the lengths of assigned portions of curves this is not necessary.; |