By A. I Markushevich
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There are other methods of approximation suitable for any curvilinear trapezoid and leading to highly accurate results while requiring computations no more difficult than with the methods just described. We shall now consider a method named after an eminent English mathematician Thomas Simpson (1710-1761), although a method like it was suggested 75 years earlier by his countryman James Gregory (1638-1675). The main idea is to replace the arcs of the graph of the function, not by chords, as was done in the case of rectilinear trapezoids (see Fig.
Now we shall prove formula (4) without expressing the coefficients a, band c in terms of the coordinates of the points A, Band C (in fact, we did not give these calculations in Section 1). We shall simply make sure that the formula is valid. The reader is asked to accept on trust that the reasoning of the proof is correct; otherwise some cumbersome computations would be necessary to find the expressions for Q, b and c. First we express the area S, which we can term the area of a parabolic trapezoid, as an integral.
Consequently, for x = 4m and a high value of m the rising straight path will be considerably higher than the logarithmic slope (see Fig. 3Ib). It is remarkable that the logarithmic slope has a rounded shape, without any irregularities, and is convex throughout. This property can be expressed in geometric terms: every arc of the graph of the logarithm lies above the chord of that arc (Fig. 32). Denoting the abscissas of the end points of an arbitrary arc y JC Fig. 32 L 1L2 by Xl XIX2 +2 X2 we can make sure that for the mean value of • a point on the arc L must indeed lie above the corresponding midpoint of the chord M.
Areas and Logarithms by A. I Markushevich