. An elementary course of infinitesimal calculus . ,yif.,yn within the respective intervals (1). But ifwe introduce the condition that none of these intervals is toexceed some assigned magnitude k, then in certain cases,which include all the types of function ordinarily met within the applications of the Calculus (and more), the value ofS will tend, as It is diminished, to some definite limitingvalue S, in the sense that by taking k small enough we canensure that S shall differ from S by less than any assignedmagnitude, however small. If the function <^ {x) admit of graphical representation
. An elementary course of infinitesimal calculus . ,yif.,yn within the respective intervals (1). But ifwe introduce the condition that none of these intervals is toexceed some assigned magnitude k, then in certain cases,which include all the types of function ordinarily met within the applications of the Calculus (and more), the value ofS will tend, as It is diminished, to some definite limitingvalue S, in the sense that by taking k small enough we canensure that S shall differ from S by less than any assignedmagnitude, however small. If the function <^ {x) admit of graphical representation, S willbe represented by the sum of a series of rectangles, whose bases 86] DEFINITE INTEGEALS. 209 ^, Aj, A„ make up the range h — a, and whose altitudes are ordinates of the curve at arbitrarily chosen points in these the limiting value S, to which S tends as the breadths ofthe rectangles are indefinitely diminished, is known as the areaincluded between the curve, the axis of x, and the extreme or-dinates x = a, x=h. See Pig. Fig. 46. The Slim which we have denoted by 2 is more fullyexpressed by tlyhx ov tli>{x)icc (4), 8a; standing for the increments h^, h^, h^ of x. The limiting value (when it exists) to which this sum converges,as the increments 8* are all indefinitely diminished, andtheir number in consequence indefinitely increased, is calledthe definite integral of the function {x) between thelimits a, and b*, and is denoted by I ydx or I {x)dx (5), J a J a • It is a little unfortunate that the word limit has to be used inseveral different senses. The word terminus might perhaps be substitutedin the present case. 210 INFINITESIMAL CALCULUS. [CH. VI the object of this notation being to recall the steps by whichthe limiting value was approached*. The connection of the notation (5) with the notation for anindefinite integral used in the preceding chapter has of courseyet to be explained. See Art. 92. Problems in which we require the limiting value
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