. Graphical and mechanical computation . ative we may use the methods of the Differential Cal-culus if the function is analytically known. Otherwise we are forced touse approximate methods — numerical, graphical, or mechanical. It is our purpose, in the following sections, to develop some of thenumerical, graphical, and mechanical methods used in approximate in-tegration and differentiation. 101. Rectangular, Trapezoidal, Simpsons, and Durands rules.—Suppose we wish to find the approximate area bounded by the curvey ~ /(*)> tne x-axis, and the ordinates x = x0 and x = xn (Fig. 101). 224 Art
. Graphical and mechanical computation . ative we may use the methods of the Differential Cal-culus if the function is analytically known. Otherwise we are forced touse approximate methods — numerical, graphical, or mechanical. It is our purpose, in the following sections, to develop some of thenumerical, graphical, and mechanical methods used in approximate in-tegration and differentiation. 101. Rectangular, Trapezoidal, Simpsons, and Durands rules.—Suppose we wish to find the approximate area bounded by the curvey ~ /(*)> tne x-axis, and the ordinates x = x0 and x = xn (Fig. 101). 224 Art. ioi TRAPEZOIDAL, SIMPSONS, AND DURANDS RULES 225 We divide the interval from x = x0 to x = xn into n equal intervals ofwidth h, and measure the (n + 1) ordinates y0, yu yi, • • • , yn-i, yn- (1) Rectangular rule. — If, starting at P0, we draw segments parallelto the x-axis through the points P0, Pi, P2, ? • • , Pn-u the area enclosedby the rectangles thus formed is given by AR = h (j0 + y\ + y* + • • • + yn-i).. ^n-a ^n-l ^n JL Fig. ioi. If, starting at Pn, we draw segments parallel to the x-axis through thepoints Pn, P„_i, . . , P2, Pi, the area enclosed by the rectangles thusformed is given by Ar = h (yi + y2 + ^3 + • • • + yn). It is evident that the smaller the interval h, the better the approxima-tion to the required area. (2) Trapezoidal rule. — If the chords P0Pi, P1P2, . . , Pn-\Pn aredrawn, then the area enclosed by the trapezoids thus formed is hi + M , , u fyn-i + yn = h [i (yo + yn) + y% + y2 4- )+ + ^ (fc^ 2+ y»-i]. This expression for the area is the average of the two expressions givenby the rectangular rules. It is evident that the smaller the interval hand the flatter the curve, the better the approximation to the requiredarea. If the curve is steep at either end or anywhere within the interval,the rule may be modified by subdividing the smaller interval into 2or 4 parts; thus, subdividing the steep interval between xn-i and x
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