. Graphical and mechanical computation . Fig. 105/. the small areas bounded by this vertical, the arc, and the horizontalsthrough A0 and Ai, equal. Proceed similarly for the succeeding construct the integral curve of the stepped line by the methodexplained above. Choose a point 5 at a convenient distance a to the leftof O and join S with the points Co, G, C2, . . , in which the extended. Fig. iosg. horizontals cut the y-axis. Then, starting at Bo, draw a line through Boparallel to SCo until it cuts the first vertical; through this point draw aline parallel to SCi until it cuts the se
. Graphical and mechanical computation . Fig. 105/. the small areas bounded by this vertical, the arc, and the horizontalsthrough A0 and Ai, equal. Proceed similarly for the succeeding construct the integral curve of the stepped line by the methodexplained above. Choose a point 5 at a convenient distance a to the leftof O and join S with the points Co, G, C2, . . , in which the extended. Fig. iosg. horizontals cut the y-axis. Then, starting at Bo, draw a line through Boparallel to SCo until it cuts the first vertical; through this point draw aline parallel to SCi until it cuts the second vertical, etc. The pointswhere the resulting broken line cuts the ordinates at A0, Ax, A2, . . ,, the Domts Bo, Bu B2, . . , are points on the required integral curve;for at each of the points A0, A\, A2, . . , the area under the curve from 242 APPROXIMATE INTEGRATION AND DIFFERENTIATION Chap. IX Ao to that point is equal to the area under the stepped line; so that asmooth curve through the points B0, B\, B2, . . will be the requiredintegral curve. Since y = - \ y dx, therefore, -—? = - y, so that the slope of thea J dx a integral curve at any point is proportional to the ordinate of the deriva-tive curve at the corresponding point. Furthermore, by the construc-tion, the slopes of the oblique lines through B0, B\, Bo, . . are propor-tional to the ordinates y0, yi, yz
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