Trapezoid rule calculations

TABLE 12.1 Trapezoidal Rule Calculations Illustrative Example 12.3... 

Using the trapezoid rule calculate the integral of the quadratic equation - 5x + 7 between x = 1 and X = 3 (check your answer against the exact answer, given by symbolic or analytical solution). 

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. 

The area under the PCP concentration-time curve (AUC) from the time of antibody administration to the last measured concentration (Cn) was determined by the trapezoidal rule. The remaining area from Cn to time infinity was calculated by dividing Cn by the terminal elimination rate constant. By using dose, AUC, and the terminal elimination rate constant, we were able to calculate the terminal elimination half-life, systemic clearance, and the volume of distribution. Renal clearance was determined from the total amount of PCP appearing in the urine, divided by AUC. Unbound clearances were calculated based on unbound concentrations of PCP. The control values are from studies performed in our laboratory on dogs administered similar radioactive doses (i.e., 2.4 to 6.5 pg of PCP) (Woodworth et al., in press). Only one of the dogs (dog C) was used in both studies. 

Table 6 illustrates the steps involved in carrying out the Wagner-Nelson calculation. The third column (f 0 Cp dt) shows the area under the Cp versus time curve calculated sequentially from t = 0 to each of the time points using the trapezoidal rule (see Sec. VIII.A). The fourth column (kei f 0 Cp di) shows each of the preceding areas multiplied by k.ei (as estimated from the tail )... 

It should be readily apparent that the trapezoidal rule does not measure AUC exactly. However, it is accurate enough for most bioavailability calculations, and the segments are chosen on the basis of the time intervals at which plasma was collected. 

Table 7 Calculation of Area Under the Plasma Concentration Versus Time Curve (AUC) Using the Trapezoidal Rule...

It is not necessary to apply the trapezoidal rule to the entire plasma concentration versus time curve in order to calculate the total AUC. After the semilog plot becomes a straight line, the remaining area out to t = can be calculated using the following equation ... 

Computer Methods These methods are easily programmed in a spreadsheet program such as Microsoft Excel. In MATLAB, the trapezoid rule can be calculated by using the command trapz(x,y), where x is a vector of x values and y is a vector of values /(%.). Alternatively, use the commands... 

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 ... 

We describe two simple ways in which discrete data may be treated to obtain the required areas (1) use of the data in histogram form, and (2) use of the trapezoid rule. These are illustrated in Figures 19.6 and. 7, respectively, in which 10 data points are plotted to represent a response curve. The curve drawn in each case is unnecessary for the calculations, but is included to indicate features of the approximations used. 

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

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. 

Eqs (1) and (2) were used to relate k and x to T before integrating by the trapezoidal rule. The table gives the intermediate calculations. 

The AUC from Figure 3.3 can be calculated as shown in Table 3.1. However, in a number of instances, the decrease in concentration is not linear but rather exponential. A more accurate method for calculating AUC is to use a log trapezoidal rule. Two consecutive observations on the exponential curve C(i ) and C(i +i) at times tj and f(i+i) are related to each other by... 

Blood samples were taken at 0.25, 0.5, 1.5, 2.5, 4.5, 6.5, 8.5, 10.5 and 12.5 h and the concentration of theophylline in serum was assayed by a spectrophotometric method [7]. The samples were analysed in duplicate. Bioavailability was calculated from the area under the concentration curve following the trapezoidal rule. 

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 ... 

Substitute Eq (2) into (1) and the result into (3). The calculations are tabulated. The trapezoidal rule was used. 

Note. Glu is the decrease in the plasma glucose concentration calculated from the area under the decreased plasma glucose level versus time curve from 0 to 6 h after rectal administration of hollow-type suppository [59] using the linear trapezoidal rule. Each value represents the mean + SE for 3-5 experiments. 

In order to obtain the degree of cure and rate of curing, we must first measure the reaction. This is typically done using a differential scanning calorimeter (DSC) as explained in Chapter 2. Typically, several dynamic tests are performed, where the temperature is increased at a constant rate and heat release rate (Q) is measured until the conversion is finished. To obtain Qt we must calculate the area under the curve Q versus t. Figure 7.17 shows four dynamic tests for a liquid silicone rubber at heating rates of 10, 5, 2.5 and 1 K/min. The trapezoidal rule was used to integrate the four curves. As expected, the total heat Qt is the same (more or less) for all four tests. This is to be expected, since each curve was represented with approximately 400 data points. 

Let us first use the trapezoidal rule to calculate the above integral... 

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... 

Solution The calculation is shown in the following table. The integral in Eq. (36) of Chapter 6 is evaluated stepwise, using the trapezoidal rule. 

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