I. AREA OF TRAPEZETTE IN TERMS OF ORDINATES 7. A trapezette is a figure of the kind with which we are familiar as the graph of a continuously varying positive function. It is bounded by a base, two sides, called the bounding ordinates, and an upper boundary which is a curved line; if this line meets the base at either extremity of the figure, the bounding ordinate is zero. Any line drawn from and at right angles to the base to meet the upper boundary is an ordinate of the figure. The figure may be regarded as traced out by an ordinate which moves from one extremity of the base to the other extremity.
8. moving ordinate of the trapezette will be denoted by u, and the abscissa of this ordinate, i.e., the distance of its foot from a certain fixed point or origin 0 on the base (or the base produced), will be denoted by x, so that u is some f unc tion of x. The breadth of the trapezette, i.e., the distance be tween its bounding ordinates, will be denoted by H. The mid ordinate is the ordinate mid-way between the bounding ordinates.
The data of a trapezette are usually its breadth and either the bounding ordinates or the mid-ordinates of a series of minor trapezettes or strips into which it is divided by ordinates at equal distances. If there are m of these strips, and if the breadth of each is h, so that H = mh, it is convenient to write x in the form +Oh, and to denote it by the corresponding value of u being The data are then either the bounding ordinates 242, • • • , of the strips or their mid-ordinates . . . , The central ordinate is the ordinate through the centroid (see §3) of the trapezette. Its distance from any straight line parallel to the ordinate is equal to the mean distance of the trapezette from the line.
9. Types of Formula.—The formulae that we have to con sider are of five types. In the first four the ordinates are supposed to be at equal distances.
(i.) The two trapezoidal rules.
(ii.) Rules such as Simpson's, in which more weight is given to some ordinates than to others.
(iii.) The rules (i.) with corrections depending on the extreme values.
(iv.) The rules (ii.) with corrections depending on the extreme values.
(v.) Formulae which involve ordinates taken at unequal in tervals.
The formulae of types (i.) and (ii.) are called "rules." This name is given to a formula which is made up of repetitions of a simpler formula. Suppose, for instance, that m is even and = 21/, so that the m strips of the trapezette can be grouped in pairs. Then, as will be seen presently, Simpson's formula gives as the area of the first pair; taking this for each pair, we get Simpson's rule denoting approximate equality): from one.of them, is of the form Then, as is easily .seen by drawing a graph of the areas of cross-sections, Simpson's formula applies to the volume of the solid; so that, if the areas of the ends are and S2, and the area of the mid section is the volume is where H is the total breadth. This is called the prismoidal formula; it applies not
only to prismoids but also to the cone, the sphere and the ellipsoid.
14. The Euler-Maclaurin Theorem.—For further progress, we have to use the Euler-Maclaurin theorem, discovered in dependently by Euler and by Maclaurin. The principle of this important theorem is that the difference between the trapezoidal area and the true area of the strip to can be expressed as the difference of the values, for u= and u= ui, of a function which only involves derivatives of u; i.e., that I 2. A Method of Construction.—The formulae given in the preceding section are not all obtained in the same way. They could all be found by a modification of the method mentioned later in §16. But the first three were originally obtained by making certain suppositions as to the upper boundary. For (a) two strips were taken together, and the top was supposed to be a parabola passing through the tops of three ordinates; for (b) three strips were taken, and the curve was supposed to be of the third degree; and for (c) four strips were taken, and the curve was supposed to be of the fourth degree.
For detailed consideration, (a) will be sufficient. We can use fig. 2; KA, MC, LB being the three ordinates u, The curve being a parabola, the tangent RCT is parallel to the chord AB. The area of the trapezium AKLB is and that of RKLT is The true area AKLBCA exceeds the former by the segment ACB, and falls short of the latter by the small pieces (spandrils) RAC and CBT. But we know that, for a parabola, the former difference is double the latter. We must therefore take a weighted mean of the two expressions, in the ratio of i : 2; i.e., the area is We could, of course, have obtained this result analytically. If u= the area is P(x2 — xo) — xe) ; and this can be reduced to the above form by using the relations ui= etc., etc., The general formula, when u is a polynomial in x of degree k, is: 13. The Prismoidal Formula.—An important application of quadrature-formulae is to finding the volume of a solid figure in terms of the areas of parallel cross-sections. In particular, sup pose that a solid is bounded by two parallel planes, and that the area of the section by a plane parallel to these, at distance x 19. Type (v.) : Unequal Intervals.—Under Type (v.) come two classes of cases.
(i.) When the given u's are not at equal intervals, the area is to be obtained by expressing the variable u in terms of x by Lagrange's formula (INTERPOLATION) and integrating.
(ii.) If we are free to choose the u's, we may want to choose them to satisfy some condition; e.g., that we should restrict ourselves to a few ordinates, and place them so as to get the most accurate value possible for the area; or that the coefficients of the u's should all be equal. Under this head come certain formu lae due to Gauss and Chebyshef (Tchebychev).