Integration trapezoidal rule

In the same way we find the value of amethane for all of the other pressures, and plot them vs. pressure, as shown in Figure 7.10. From this plot we find the integral we need by numerical integration (trapezoid rule) as 290 psi ft /lbmol. Thus, for methane... 

Several related rorles or algorithms for numerical integration (rectangular mle, trapezoidal rule, etc.) are described in applied mathematics books, but we shall rely on Simpson s mle. This method can be shown to be superior to the simpler rules for well-behaved functions that occur commonly in chemistry, both functions for which the analytical form is not known and those that exist in analytical form but are not integrable. 

This method approximates/(x) by a parabola on each subintei val. This rule is generally more accurate than the trapezoidal rule. It is the most widely used integration formula. 

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. 

Improper integrals of the other types whose problems involve both limits are handled by open formulas that do not require the integrand to be evaluated at its endpoints. One such formula, the extended midpoint rule, is accurate to the same order as the extended trapezoidal rule and is used when the limits of integration are located halfway between tabulated abscissas ... 

Semi-open formulas are used when the problem exists at only one limit. At the closed end of the integration, the weights from the standard closed-type formulas are used and at the open end, the weights from open formulas are used. (Weights for closed and open formulas of various orders of error may be found in standard numerical methods texts.) Given a closed extended trapezoidal rule of one order higher than the preceding formula, i.e.. 

Two simple numerical methods are often used to determine the area under the curve that equals the desired integral. They involve the use of the trapezoidal rule and Simpson s rule. 

Figure A1.5 Graphical integrations using (a) the trapezoidal rule and (b) Simpson s rule.

Another approach is to use Simpson s rule instead of the trapezoidal rule for the integration. With Simpson s rule, three successive points, e.g,, n), are... 

Euler s theorem 612 exact differentials 604-5 extensive variables 598 graphical integrations 613-15 Simpson s rule 614-15 trapezoidal rule 613-14 inexact differentials 604-5 intensive variables 598 line integrals 605-8... 

The hnal step in the design calculations for a laminar flow reactor is determination of mixing-cup averages based on Equation (8.4). The trapezoidal rule is recommended for this numerical integration because it is easy to implement and because it converges O(Ar ) in keeping with the rest of the calculations. 

Both F 0) and F R) vanish for a velocity profile with zero slip at the wall. The mixing-cup average is determined when the integral of F(r) is normalized by Q = 7tR u. There is merit in using the trapezoidal rule to calculate Q by integrating dQ = InrVzdr. Errors tend to cancel when the ratio is taken. 

This modified density Is a more slowly varying function of x than the density. The domain of Interest, 0 < x < h, Is discretized uniformly and the trapezoidal rule Is used to evaluate the Integrals In Equations 8 and 9. This results In a system of nonlinear, coupled, algebraic equations for the nodal values of n and n. Newton s method Is used to solve for n and n simultaneously. The domain Is discretized finely enough so that the solution changes negligibly with further refinement. A mesh size of 0.05a was adopted In our calculations. 

Similarly we can estimate the specific secretion rate. It is obvious from the previous analysis that an accurate estimation of the average specific rates can only be done if the integral Jxvdt is estimated accurately. If measurements of biomass or cell concentrations have been taken very frequently, simple use of the trapezoid rule for the computation of Jxvdt may suffice. If however the measurements are very noisy or they have been infrequently collected, the data must be first smoothed through polynomial fitting and then the integrals can be obtained analytically using the fitted polynomial. 

The correction to this expression involves multiple derivatives, although the basic equation, Eq. (60), does not The development of tins result, as above for the trapezoid rule, leads to the relation for the integral over the range a to b in the form... 

The integration was carried out with the extended trapezoidal rule for an integral over function fix)... 

The trapezoid rule [see Numerical Integration (Quadrature) ] is applied to obtain... 

In order to study the implications of Equation 2, it was evaluated at 80 points in the range xl = 0 to it. At xl 0, L Hospital s rule from calculus was needed. For larger xl, Equation 3 was evaluated for each xl using trapezoid rule numerical integration, yielding values for use in Equation 2. It was found that the rate of deposition is the highest for xl near zero, diminishing to zero at xl = ir. ... 

Equations (A), (B) and (C) are used in the algorithm to obtain the information required. Step (3) is used to calculate kA from equation (B), and step (4) is not required. Results are summarized in Table 12.2, for the arbitrary step-size in fA indicated G = tl[kA 1 -/A)j, and G represents the average of two consecutive values of G. The last column lists the time required to achieve the corresponding conversion in the second column. These times were obtained as approximations for the value of the integral in equation (A) by means of the trapezoidal rule ... 

These equations, (A) to (F ), may be solved using the E-Z Solve software or the trapezoidal rule for evaluation of the integral in (A). In the latter case, the following algorithm... 

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. 

Softwares for numerical integration of equations include the calculator HP-32SII, POLYMATH, CONSTANTINIDES AND CHAPRA CANALE. The last of these also can handle tabular data with variable spacing. POLYMATH fits a polynomial to the tabular data and then integrates. A comparison is made in problem PI.03.03 of the integration of an equation by the trapezoidal and Runge-Kutta rules. One hundred intervals with the trapezoidal rule takes little time and the result is usually accurate enough, so it is often convenient to standardize on this number. 

For manual integration of tabular data, the trapezoidal or Simpson s rules usually are adequate, Between two points the trapezoidal rule is... 

The available software for numerical integration of first order ODEs is applicable only when dC/dt is available explicitly. Here a "root solver" is used to find the relation between C and r. Then the relation to t is obtained by integration with the trapezoidal rule,... 

For comparison, integration with the trapezoidal rule with 100 intervals gives k = 0.0716. 

The integration is done numerically, trapezoidal rule, 200 intervals. The almost constant results of column 4 confirm the rate equation and have an average value... 

Integration also is done with the trapezoidal rule, 100 intervals, and the results are obtained. 

The third equation can be integrated with the trapezoidal rule. Values are shown in the last column and have the average shown. 

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