. An elementary course of infinitesimal calculus . 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 t
. An elementary course of infinitesimal calculus . 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 = Fig. 49. 87-88] DEFINITE INTEGRALS. 215 Take Aj = Aj = = A„, =h, say, so that nh^b — a, and let y^, y^, y„, be the values which the function to be integrated has at the beginnings of the successive intervals, viz. 2/i = «. yi=a + h, y^ = a + 2h, y„ = a+{n-l)h. Then % = ah+ {a + h)h + {a + 2h) h+ ... + {a + {n~\)h]h = nah+{\ + 2+ ... + {n- l)}h^ = nah + ^{n- l)!i? = a{b-a) + i(l-lyb-af. When h = 0, we have n = co , and the limiting value of the aboveexpression is a{b-a) + ^(b-aY=^{b° - a),rb Hence I xdx--=^(b^-a) (2). Ja Tofind I a?dx (3). Ja This is equivalent to finding the area included between theparabola y = a?, the axis of x, and the ordinates x = a,x = h.
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