Trapezoid rule examples

Example Evaluate the same integral (3-75) using the trapezoid rule and then apply the Romberg method. To achieve four-digit accuracy, any result from J9 through Li are suitable, even though the base results (Z1-Z4) are not that close. 

For numerical evaluation, we use the simple trapezoidal rule, and a stepwise procedure similar to that in Example 12-5, which can be readily implemented by a spreadsheet program. For convenience, in equation (C), we let... 

Table 19.1 Spreadsheet calculations for Example 19-1, using the trapezoid rule...

Values of t and of obtained using the trapezoidal rule are 21.9 s and 161.4 s2, respectively. These values are used in Examples 19-8 and -10 below. 

Evaluate the integral in equation 21.6-5 by means of the E-Z Solve software or an approximation such as the trapezoidal rule, as in Example 21-5. 

Find the drying time by integration of the reciprocal rate as in Example 9.3, with the trapezoidal rule. 

Integration is accomplished numerically with the Simpson or trapezoidal rules. This method is applied in Example 11.2. 

The relation between interfacial and bulk concentrations is that of Eq. (13.157), (y -y)/(x -x) = -kL/kG. At a series of values of x, corresponding values of y and y may be read off with the graphical constructions shown on Figures (b) and (c) of this example. The values for slope = — 1 are tabulated, but those for slope = oo are calculated from the equations of the equilibrium and operating lines and are not recorded. The integrands of Eq. (13.160) also are tabulated for both cases, and the numbers of transfer units are obtained by integration with the trapezoidal rule ... 

Solution For a first-order reaction, we can arbitrarily set am = 1 so that the results can be interpreted as the fraction unreacted. The choices for 7 and J determined in Example 8.4 will be used. The marching-ahead procedure uses Equations (8.25), (8.26), and (8.27) to calculate concentrations. The trapezoidal rule is used to calculate the mixing-cup average at the end of the reactor. The results are... 

Such a normalization condition can be readily discretized by considering, for example, the simple trapezoidal rule for performing the numerical quadrature ... 

The errors resulting from the use of Simpson s one-third rule are much smaller than those associated with the use of the trapezoidal rule. This is illustrated by the example given in Fig. 1. The points shown in this figure were generated from the function j = 10 - + 0.03/ and are given below ... 

For the transfer unit determination with the given ratio of mass transfer coefficients, corresponding values of (y, y ) are found by intersections of the material balance and equilibrium lines with lines whose slopes are -kjk = -1 as indicated on Figure (a) and in detail with Example 13.12. These values are tabulated together with the couesponding integrands. The number of transfer units is found by trapezoidal rule integration of... 

Figure (a) and in detail with Example 13.12. These values are tabulated together with the corresponding integrands. The number of transfer units is found by trapezoidal rule integration of... 

Note that the trapezoidal rule is the first-order member of this method. In the above example, fourth-order Newton-Cotes integration for /x 0.9 yields 71=1.00, tr= 26.67, gtr= 24.00, ar= 1.72, and Np = 24.00. In fact, almost equally accurate results can already be obtained with a second-order Newton-Cotes fit. 

In this example, F(0) = 0 because r = 0 and F(R) = 0 because V iR) = 0. The mixing-cup average is determined when the integral of F(r) is normalized by 2 = There is merit in using the trapezoidal rule to calculate Q = AcM by integrating dQ = 2jtr dr. Numerical errors of integration tend to cancel when the ratio is taken. 

Code for Example 8.2 illustrates the use of the trapezoidal rule for evaluating both the numerator and denominator in this equation. The results are as follows ... 

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