Trapezium rule

Evaluation of a one-dimensional integral using the trapezium rule. The area under the curve is approximated mm of the areas of the trapeziums. 

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. 

For small samples we can integrate the global interpolating polynomial. For larger samples the trapezium rule... 

The model may be conveniently fitted to experimental data using a straightforward spreadsheet. Suppose that we have a set of TG data in the form of three columns—time (s), temperature (K), and mass/initial mass. The basic idea is to assume values for the Arrhenius parameters /= In A and Tk for some fixed value of mv. We then calculate k and hence x, using the fact that x = l 0k(T(t ))dt (a suitable numerical rule such as the trapezium rule suffices for this). Having obtained x, we may... 

In the Simpson s rule procedure, a similar division of the area, defined between the bounds of the integration and the curve of the function is made. However, for Simpson s rule, it is necessary to choose an even number of intervals. Then, in the same notation as for the Trapezium rule procedure. 

For example, the trapezoid rule (also known as the trapezoidal rule or trapezium rule) is close ... 

The integral of a continuous function, f x), may be approximated numerically using the trapezium rule [10]. The function is divided into a series of discrete intervals (discretised) and evaluated at each point, x. The area imder the curve of a single interval is approximated as being trapezoidal, and the area of all such trapezoids is summed to give the integral. In general for a function discretised into N points. 

From equation (6.20) and trapezium rule integration, the filtration time ... 

The fractional solute recovery (F) is determined from the (p vs. W dataset calculated in (4). Using equation (6.122), F is calculated by trapezium rule integration where the upper limit is defined by the specified wash ratio ... 

This system can be solved with the implicit method of Crank-Nicholson (trapezium rule) this... 

An alternative method is to apply the trapezium rule, where the integral is treated as series of trapeziums. 

Thermodynamic integration interpolates between these two extremes. In the limit of linear solvent response, the trapezium rule is exact, and the integral is the mean of the end points. The linear response approximation for the oxidation free energy is therefore determined by the IP of the reduced state and EA of the oxidized state" " ... 

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