Quadratures, METHOD OF.

Quadratures, METHOD OF. This name is applied to any arithmetical method of determining the area of a curve. When the exact area is known a square whose area is equal to it can be found—hence the term 'quadratures.'

It has been shown, under the heading CALCULUS, that the area of a curve whose equation is $y\; =\; f(x)$ is $\backslash int\; y\; dx$, and can therefore be found when the integral can be evaluated. Hence the approximate determination of the value of a definite integral is obtainable by the method of quadratures.

Let it be required to find the area bounded by a portion of a curve, the ordinates at its extremities, and the axis. The usual method of procedure is to divide the portion of the axis which is included between the two ordinates into a number of equal parts, and to erect ordinates at the points so obtained. The area is approximately equal to the product of one of the given equal parts into half of the sum of the two extreme ordinates together with the sum of all the intermediate ordinates. To obtain a very accurate result by this process the number of equidistant ordinates must be so great that the portions of the curve which are intercepted by successive ordinates are very nearly straight.

A better method, due to Simpson, consists in drawing, through the first, second, and third points obtained as above on the curve, a parabola whose axis is parallel to the ordinates, and repeating this process with the third, fourth, and fifth points, and so on—the points being chosen so that the total number of points is even. The area of the given curve will be approximately equal to the sum of the areas of the various portions of the parabolas included between successive ordinates when these ordinates are sufficiently close together. It is therefore approximately equal to one-third of the product of one of the given equal portions of the axis into the sum of the extreme ordinates together with twice the sum of all the odd intermediate ordinates and four times the sum of all the even intermediate ordinates.

When the successive equidistant ordinates are very close together, the area is approximately equal to the product of the common intercept on the axis between successive ordinates into the sum of all the ordinates. The labour involved in the estimation of an area by this process would be fatal to its employment unless the number of ordinates was small. But, if the ordinates were few in number, considerable error would in general result unless a correction could be applied. This method is adopted in that process which is known as the method of quadratures par excellence, and which is as follows: Let $y\_0,\; y\_1,\; \backslash dots,\; y\_n$ be the several equidistant ordinates, and let $a$ be the intercept on the axis between $y\_0$ and $y\_n$. Also let $s$ be the sum above referred to; and let $\backslash Delta\; y\_0\; =\; y\_1\; -\; y\_0$, $\backslash Delta\; y\_1\; =\; y\_2\; -\; y\_1$, &c.; $\backslash Delta^2\; y\_0\; =\; \backslash Delta\; y\_1\; -\; \backslash Delta\; y\_0$, $\backslash Delta^2\; y\_1\; =\; \backslash Delta\; y\_2\; -\; \backslash Delta\; y\_1$, &c.; and so on. The value of the whole area is (not $s$, but)

It will not in general be necessary to proceed beyond the fifth difference. As an example we shall find the area of the curve $y\; =\; x^3$ between the limits $x\; =\; 10$ and $x\; =\; 15$. In this case all differences beyond the third vanish, and $a/n\; =\; 0.5$ if we make eleven ordinates in all. The following table represents the results:

Hence we have $s\; =\; 22515.625\; \backslash times\; 0.5\; =\; 11257.8125$; $\backslash frac\{1\}\{2\}\; \backslash times\; 0.5\; \backslash times\; (y\_\{10\}\; +\; y\_0)\; =\; 1093.75$; $\backslash frac\{1\}\{2\}\; \backslash times\; 0.5\; \backslash times\; (\backslash Delta\; y\_0\; -\; \backslash Delta\; y\_\{10\})\; =\; 6.61458$; $\backslash frac\{1\}\{2\}\; \backslash times\; 0.5\; \backslash times\; (\backslash Delta^2\; y\_0\; +\; \backslash Delta^2\; y\_\{10\})\; =\; 0.78125$. We therefore get by this method, approximately, for the value of the area the quantity 10156.67. The correct value is 10156.25, and so the error is less than one in twenty thousand.

This method is of extreme utility in the evaluation of definite integrals when rigorous processes are not attainable.