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112.] If however the sum of the series in the right-hand member of (202) cannot be expressed in general terms of n, yet an approximation may be made to it, and may be carried to any extent. For this purpose let the range be divided into a finite number of equal parts; say, into n equal parts; and let the value of the element-functions corresponding to the commencement of each part of the range be calculated; then the product of the sum of all these element-functions and of the element of the range will be the approximate value of the definite integral; and the larger n is, the nearer will the value thus determined be to the true value of the integral. This process may of course involve long and intricate calculations; yet the process is theoretically perfect, and may always be applied. A small value of n will be sufficient for a first approximation, and this is frequently of considerable practical use.

For an example of the method let us take the definite integral s dx

· I have chosen one, the value of which is known, that the approximate results may be compared with the true result.

Let the range, which = 1, be divided into four equal parts, each of which = á: then

n dix 1 16 4 16)
Jo 1+ 22 = 7 21 +17+ 5 + 25}

= .25 +.234+.2+.16

= .844. Again, let the range be divided into ten equal parts, each of which = 10; then = to {1 + 101 + 104 + 109 + 16 + 125 + 136

100 100 1007

+ 149 + 164 + 1815 = {1+.99+.9615+.9174+.862+.87.7352

+.6711 +.6097 +.5524} = .80993.

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1.17

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errors in the two preceding calculations are respectively +.059 and +.02454.

113.] Now let the reader refer to the geometrical interpretation of integration given in Art. 3, and compare with it the process of the preceding Article. Let y = f'(x) be the equation to the plane curve represented in fig. 16; let om, = X, OM = x, OM, = Xn; MN = dx, MP = y = f'(x): so that f'(x) dx expresses the area of MPQN, when Mn = dx = an infinitesimal. Evidently therefore will F'(x) dx express the area contained between the curve, the extreme ordinates M, P, and M, Pn, and the x-axis, Under these circumstances the range M, M, is divided into an infinite number of elements. If however the number of parts into which the range is divided is finite, and the sum of the elementfunctions corresponding to these is calculated, that sum will be only an approximate expression for the value of the area. For suppose mn, in fig. 16, to be a part of the range, = i, say; then

F'(x) = MNXMP = the rectangle MPRN; which is short of the corresponding part of the curvilinear area by the area of PRQ: and as a similar result is true of each part of the range, the sum of all the rectangles will be less than the required area by the sum of all the similar triangular pieces. If the bounding curve approaches the axis of x, as x increases, the sum of the areas corresponding to the finite partition of the range of the x-integration will be greater than the true result by the sum of similar triangles. The differences however between the true results and the approximate results thus determined will be less, according as the number of parts into which the range is divided is greater.

The process of thus approximating to the value of a curvilinear area by the geometrical expression of the mode of approximate integration explained in the preceding Article is so exact, that the latter has been called Approximate Integration by summation of ordinates at equal finite intervals.

This geometrical illustration suggests a more exact process of approximation. It is plain that the product of an into the semi-sum of me and nQ is nearer to the true value of the area MNQP than MN XMP. So that if i represents the required area, or definite integral, and if yo, yv, Y, ... Yn denote the ordinates, or the element-functions, corresponding to X, Xo+i, ... Xm, then

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Of this Theorem we have the following interesting application. Let f'(x) = log x, and let the limits of integration be m and m+n; and let us suppose i = 1; then

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= (m+n) log (m+n) m logm-n. :log m+ log (m+1) +log (m+2)+...+log (m +n-1)

= (m+n) log (m+n) –m log m-n-{log (m+n) – log m}

= (m + n-3) log (m+n)– (m – 7) log mæn; and taking numbers instead of logarithms, we have m(n+1) (m+2)... (m + n-1) = (m + n)*+*-*m-m+e-". (206)

Thus the right-hand member is an approximate value of the product of n numbers integral or fractional, in arithmetical progression, of which the common difference is unity. If m = 1, 1.2.3 ... n = (1+n)*+$e-".

(207) Now the difference between the true value and the approximate value given in (206) of m (m+1) ... (m+n-1) will become less, the larger n is, because the greater the range of integration is, the smaller proportionally becomes the difference between the successive elements, which we have assumed to be unity. Let us suppose n = 0; then since, when n = 0, (m+n)*+--+ = n*+--+(1+m)" (1 + m)***

= nm+n-dem;

therefore from (206),

m (m +1) (m+2)... (m+n-1) = m-9+cm +n-te-", (208) when n = 0.

This is indeed only an approximate value for the product of the factorials, but we shall hereafter be able to correct it for the particular case given in (208), viz. when n = 0.

Again, if (205) is applied to the Example in Art. 112, and the range is divided into ten equal parts, the result = 7.8493, which is less than the true value by only .0046, and is a much more exact approximation than those found in the preceding Article.

114.] The partition of the range of integration into equal finite parts and the calculation of the element-function corresponding to each part, produces other formulæ for the determination of approximate values of definite integrals; and as these are also of considerable practical use in Mensuration it is necessary to demonstrate them.

Let the element-function be f(x); and let the range x n - %, be divided into n equal parts, each of which=i; so that Xn-Xo=ni.

Also let the difference between F(x+i) and F(x), which is generally finite, since i is finite, be denoted by Af(x); so that we have F(x+i) = F(x) + AF(x);

(209) in reference to which it may be observed that a becomes d, when i becomes dx. Let the variable x receive another finite increment i; then from (209), F(x+2i) = f(x + i) + AF(x+i)

= F(x) +A F(x) + {F(x) + A F(x)}

= F(X) +2 A F(x) + AA F(x). Let an be denoted by aʼ; this substitution being merely a matter of notation, and independent of any supposition as to whether A denotes an operation subject to the index law. Similarly let and be denoted by A3; and so on; then

F(x + 2i) = f(x) + 2a F(x)+a? F(x). Again, let x receive another increment i; then

F(x+3i) = f(x) + 3 A F(x) + 3F(x) + A3 F(x); the law of the coefficients being evidently that of the Binomial Theorem. Consequently, if x is increased m times successively by i, we shall have

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+. . . . . . . . . . . . . . . . . ; (212) replacing X.-X, by ni, this becomes

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