. Applied calculus; principles and applications . rst integralcurve is the same as the number that represents the areabetween the original curve, the axis (or axes for some func-tions), and the ordinate for this same abscissa. Hence, theordinates of the first integral curve may represent the areasof the original curve bounded as stated. It may be seen also that for the same abscissa x^ the numberthat expresses the slope of the first integral curve is the sameas the number that measures the length of the ordinate of 227 228 INTEGRAL CALCULUS the original curve. Hence the ordinates of the origin
. Applied calculus; principles and applications . rst integralcurve is the same as the number that represents the areabetween the original curve, the axis (or axes for some func-tions), and the ordinate for this same abscissa. Hence, theordinates of the first integral curve may represent the areasof the original curve bounded as stated. It may be seen also that for the same abscissa x^ the numberthat expresses the slope of the first integral curve is the sameas the number that measures the length of the ordinate of 227 228 INTEGRAL CALCULUS the original curve. Hence the ordinates of the originalcurve may represent the slopes of the first integral The integral curve of the curve of equation (2) is calledthe second integral curve of the original curve of equation(1). The integral curve of the second is called the thirdintegral curve of the original curve (1), and so on. Thusfor any given curve there is a series of successive integralcurves.* The function cos 6 and the first and second integral curvesare shown with their Let y = cosd be the fundamental function, theny = I cosedd = sine -\- C,where C is zero, as y is zero when d is zero; andy = I sinede = —cosd -\- C, * The statements in the three paragraphs above with some differenceof words are given in Murrays Integral Calculus, where a fuller treat-ment will be found in the Appendix. APPLICATION TO BEAMS 229 where C is one, as y is zero when 6 is zero. Hence,y = sind and y = 1 — cos 6 = vers d are the first and second integral curves of the curve y = cos 6. It is seen that the ordinate of the first integral curve ate = 7r/2 is +1, that number being the same number thatmeasures the area under the fundamental curve for the sameabscissa; the ordinate being zero at ^ = tt indicates that thealgebraic sum of the areas of the fundamental curve is zeroand hence that the area below the axis from 6 = 7r/2 to t isexactly equal to that above from ^ = 0 to 7r/2; the ordinatebeing zero again a,t 6 = 2 w indic
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