Numerical integration rule

ENZ, AC. and MALIK, A.A., An Imbedded Family of Fully Symmetric Numerical Integration Rules, SIAM J. Numerical Analysis,Iti (3), June 1983, 580-588. 

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. 

The integral of the Gaussian distribution function does not exist in closed form over an arbitrary interval, but it is a simple matter to calculate the value of p(z) for any value of z, hence numerical integration is appropriate. Like the test function, f x) = 100 — x, the accepted value (Young, 1962) of the definite integral (1-23) is approached rapidly by Simpson s rule. We have obtained four-place accuracy or better at millisecond run time. For many applications in applied probability and statistics, four significant figures are more than can be supported by the data. 

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. 

Fig. 39.15. Area under a plasma concentration curve AUC as the sum of a truncated and an extrapolated part. The former is obtained by numerical integration (e.g. trapezium rule) between times 0 and T, the latter is computed from the parameters of a least squares fit to the exponentially decaying part of the curve (P-phase).

This parameter can be obtained by numerical integration, for example using the trapezium rule, between time 0 and the time T when the last plasma sample has been taken. The remaining tail of the curve (between T and infinity) must be estimated from an exponential model of the slowest descending part of the observed plasma curve ((3-phase) as shown in Fig. 39.15. The area under the curve AUC can thus be decomposed into a tmncated and extrapolated part ... 

The truncated part of the integral can be obtained by numerical integration (e.g. by means of the trapezium rule) of the function rCp(r) between times 0 and T. The mean residence time MRT is an important pharmacokinetic parameter, especially when a substantial fraction of the drug is excreted or metabolized during its first pass through an organ, such as the liver. 

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. ... 

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. 

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,... 

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... 

The last column of the table records the numerical integration by the trapezoidal rule with 100 intervals and provides a good check of the experimental data. 

This scheme has tight connections with the Simpson s rule for numerical integration. 

On a numerical level, the integrals in Eq. (9) are substituted by numerical integration, e.g., by means of the trapezoidal rule ... 

As a matter of fact, most mechanisms do not have explicit solutions and require numerical integration. The few mechanisms discussed so far are exceptions rather than the rule. Fortunately, numerical integration is always possible and next we demonstrate how this can be achieved. 

Numerical Integration. Using Simpson s rule, applicable to an even number of uniformly spaced intervals on the axis, we find for the data of Table E5.4,... 

The source of mass, M, could be a function of t, or could be a steady value. The solution to equation (E2.6.2) is normally found with a numerical integration routine using integration techniques such as Simpson s rule. 

By numerical integration we can use Simpson s rule to evaluate the data in columns 1 and 5 of Table 5.5. 

Using Simpson s rule or some other appropriate numerical integration technique, determine the area under the curve obtained from step (3). From the area, the required value for the tower characteristic (KaV/L) can be determined. 

A more accurate method of numerical integration is by the use of Simpson s rule. This method, however, requires that the integration range be divided into an even number of intervals of equal width h. This requires an odd number of points on the abscissa, which are numbered from 0 to n. Simpson s rule gives... 

Vickery and Taylor (81) used a Naphtali-Sandholm method containing all of the MESH equations and variables [M2C + 3) equations] with the variables represented by x. H is the Jacobian from the Naphtali-Sandholm method solution of the known problem, G(x) = 0, This is numerically integrated from t = 0 to t - 1, finding a H, at each Step and updating H when the solution is reached at each step, With Hj. and H, known, dxjdt is solved, and with step size t, a new set of values for the independent variables x is found by Euler s rule... 

---