Trapezoidal method

Figure 19.7 Basis for trapezoidal method of area determination, by equation 19.3-15, based on points marked x (curve unnecessary—drawn for illustration only)...

Two features of the trapezoidal method are that we use a uniform spacing between the positions where we evaluate fix) and that every evaluation of fix) (except the end points) is given equal weight. Neither of these conditions is necessary or even desirable. An elegant class of integration methods called Gaussian quadrature defines methods that have the form... 

Verapamil and its active metabolite norverapamil were assayed from serum (frozen at -18°C) using a gas chromatographic-mass spectrometric method [5]. The detection limit of the method was 1 ng/ml. The areas under the concentration-time curves (AUC 0-32 h) were calculated by the trapezoidal method. Statistical evaluation was carried out using the paired Wilcoxon test and paired Student s t test. 

The simplest method of numerical integration is the trapezoidal method, where the abscissa is divided into intervals and each resulting area is estimated as the abscissa interval times the average of the initial and final ordinate in the interval ... 

One of the two integrals (4.67) may be evaluated analytically the second one (elliptic) may be calculated numerically by the trapezoidal method with the introduction of a small imaginary part in (4.67) to avoid spurious oscillations due to the finite number of integration points. The resulting density-of-states function is shown in Fig. 4.9. The band is asymmetric because of the equivalent term Vx j it exhibits three van Hove singularity points two discontinuities at the boundaries, and one logarithmic divergence corres-... 

The elimination rate constant (fi) is calculated from the linear regression analysis of plasma concentration-time curve. The area under the curve (AUCo oo) is calculated using the linear trapezoidal method with extrapolation of the terminal phase to infinity (Ciast//1), where C ast is the last measured concentration. Other parameters calculated are as follows Total body clearance (Cl) = dose/AUC volume of distribution (Earea) = Cl//l and elimination half-life (11/23) = 0.693//1 (see Note 14). 

Where applicable, pharmacokinetic parameters (Cmax, Tmax, AUCo-24, hn) were calculated using a non-compartmental analysis employing a linear/log trapezoidal method. 

F. lavernaro and D. Trigiante, Discrete conservative vector fields induced by the trapezoidal method, JNAIAM J. Numer. Anal. Indust. Appl. Math., 2006, 1(1), 113-130. L. F. Shampine, P. H. Muir and H. Xu, A user-friendly fortran BVP solver, JNAIAM J. Numer. Anal. Indust. Appl. Math., 2006, 1(2), 201—217. 

A common use of numerical integration is to determine the area under a curve. We will describe three methods for determining the area under a curve the rectangle method, the trapezoid method and Simpson s method. Each involves approximating the area of each portion of the curve delineated by adjacent data points the area under the curve is the sum of these individual segments. 

Calculations at high conversion - The quantities discussed above are calculated at regular conversion increments and are integrated using the trapezoidal method as is described in several of our earlier papers (1, 3). Average values of n-add distributions, etc., are then reported for conversions specified by the user. As an option the program will also provide information about the concentrations of unreacted monomers at various conversions. 

For our absorption peak, the trapezoid method using different widths produces the following estimates for the integral ... 

In general the trapezoid method is inferior to the rectangular method. 

A more accurate method can be achieved by combining the rectangular and trapezoid methods into the technique referred to as Simpson s method, ... 

Spreadsheet Summary In the first experiment in Chapter 11 of Applications of Microsoft Excel in Analytical Chemistry, numerical integration methods are investigated. These methods are used to determine the charge required to electrolyze a reagent in a controlled-potential coulometric determination. A trapezoidal method and a Simpson s rule method are studied. From the charge, Faraday s law is used to determine the amount of analyte. 

Step 2 Determine Area under Curve (AUC0 - ) with trapezoid method... 

AUC calculated by the trapezoidal method with extrapolation of the terminal slope to infinity using linear regression. 

Figure 20-55. The Trapezoid Method to Determine Peak Areas Showing How the Template Is Used.

Pharmacokinetic Analysis. Standard noncompartmental analyses were conducted to assess ATI and ATF pharmacokinetics using WinNonlin software (v. 2.1) (Pharsight, Mountain View, CA). The areas under the plasma concentration versus time curve from time zero to inhnity (AUCint) were determined via the log-linear trapezoidal method. The terminal half-life was determined from the relationship of ti/2 = In 2/, where k is the negative slope of the terminal phase of the InC versus time plot. Systemic clearance (CL) was estimated by dividing the administered dose by AUCint. The volume of distribution at steady state (Vss) was determined by the product of clearance and the mean residence time. 

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