. An elementary course of infinitesimal calculus . (6); hence by taking k small enough we can ensure that | S — /S |shall be less than any assigned quantity, however small*. A similar proof obviously applies if the function ^(x)steadily decreases throughout the range b — a. It follows that the final result also holds when the rangeadmits of being broken up into a finite number of smallerintervals within each of which the function either steadilyincreases or steadily decreases. See Fig. 48. It has been supposed that b > a. If 6 < a, the intervalshi, hi, ..., hn will be negative, but the a
. An elementary course of infinitesimal calculus . (6); hence by taking k small enough we can ensure that | S — /S |shall be less than any assigned quantity, however small*. A similar proof obviously applies if the function ^(x)steadily decreases throughout the range b — a. It follows that the final result also holds when the rangeadmits of being broken up into a finite number of smallerintervals within each of which the function either steadilyincreases or steadily decreases. See Fig. 48. It has been supposed that b > a. If 6 < a, the intervalshi, hi, ..., hn will be negative, but the argument is sub-stantially unaltered. * The proof is a development of that given by Newton, Principia, lib. i.,sept, i., lemma iii. (1687). It would be easy to eliminate all geometricalconsiderations and present the argument in a purely quantitative form. 214 INFINITESIMAL CALCULUS. [CH. VI. Fig. 48. 88. Examples of Definite Integrals calculated ab The meaning of a definite integral may be furtherillustrated by the study of a few cases in which the limitingvalue can be calculated from first principles. The methodsto which we are obliged to have recourse for this purposewill at all events enable the student to appreciate theenormous simplification which was introduced into the sub-ject by the invention of the special rule of the IntegralCalculus, to which we afterwards proceed (Art. 92). Tofind I xdx (1). This is equivalent to finding the area of the trapezium PABQin the figure, where OA = a, OB = 6, and OPQ is the straight liney = x.
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