Trapezoidal integration

The variables in Eq (4) are separable. Some of the integrated results of P and na are tabulated. Then Vr is found with Eq (2), using trapezoidal integration. Some of those results also are shown. 

During the discharge period, reaction continues in the tank in accordance with the same rate equation, for the 20 minutes of the discharge time. The concentrations during this period are tabulated. The mean value to storage is obtained by trapezoidal integration,... 

Response to an impulse of tracer was measured with the tabulated results. The mean residence time is found by trapezoidal integration. 

The table summarizes the data and the calculations with Eqs (3), Trapezoidal integration is used. 

B. TRAPEZOIDAL INTEGRATION. Trapezoidal integration draws a straight line between the values of the input. 

Figure 8.52 IsSpice results of Colpitts oscillator (trapezoidal integration).

Data are presented as mean SEM. Tail flick latencies and paw pressure thresholds were converted to the percentage of maximal possible effect. The area under the time-effect curve was calculated by accumulating the effect measured at discrete time intervals using the trapezoidal integration method. The results were analyzed by ANOVA with repeated measures followed by Scheffe and Dunnett tests. The injury score for each technique and each solution was compared using two-way ANOVA followed by the Scheffe test. The frequency (i.e., the number of rats with lesions) in each group was analyzed by chi-square test. 

Figure 8 The area under the AAS profile using (a) rectangular and (b) trapezoidal integration...

Area under the plasma/seruin/blood concentration-time curve from time zero to time t (AUCO-z), where t is the last sampling time point with a measurable concentration of the API in the individual formulation tested. The method of calculating AUC-values should be specified. In general AUC should be calculated using the linear/log trapezoidal integration method. The exclusive use of compartmental-based parameters is not recommended. 

In cell AB3 compute the peak area A by trapezoidal integration, with the instruction = SUM(AB6 AB205). 

For large values of /j. we encountered another difficulty the number of simulated points was too small to use a simple trapezoidal integration. For example, at p, = 0.9, we had only about ten points that make a significant contribution to the peak. Even though this was a simulation, we could not increase the point density, because t = 1 is the smallest step size of the model in many practical integrations, the step size may be limited by instrumental factors and likewise be fixed. In computing the standard deviation this difficulty was exacerbated because we calculated oy2 as the difference between two larger numbers. 

When we have too few points to justify linearizing the function between adjacent points (as the trapezoidal integration does) we can use an algorithm based on a higher-order polynomial, which thereby can more faithfully represent the curvature of the function between adjacent measurement points. The Newton-Cotes method does just that for equidistant points, and is a moving polynomial method with fixed coefficients, just as the Savitzky-Golay method used for smoothing and differentation discussed in sections 8.5 and 8.8. For example, the formula for the area under the curve between x, and xn, is... 

First, we approximate the integral F k) with the trapezoidal integration rule using an equidistant grid spacing for j = — 1... 

The polymer concentration and monomer disappearance were Incremented by simple trapezoidal Integration in the axial direction. 

Z = trapz(XjY) computes the integral of Y with respect to X using trapezoidal integration. Inputs X and Y can be complex. 

Aerr is the proportionality constant [windows used. S is the amount of sampling performed at each window. This analysis assumes use of trapezoidal integration. From equation (78) it can be seen that the total error is inversely proportional to both IVw and Sw, so if we increase one, we can decrease the other by the same proportion and maintain the same overall error. That is, if the total amount of data collection /VwSw remains constant, so does the total error. 

One of the safe approach to convert Eq.s 1 and 2 to finite difference form, is trapezoidal integration from t to t+A (A is time interval). 

There was a strong positive correlation between the HOME score and the Bayley MDI. These two measures were negatively correlated with postnatal PbB concentrations (expressed as integrated PbB which is derived by trapezoidal integration of each individual s blood lead curve derived from PbB estimates of cord, 6-, 15- and 24-month samples) (Table 6). A very low HOME score also appeared to be associated with a low Bayley PDI, but no significant relationship was found between these two measures across the rest of the HOME score range. 

In this set of equations, the +1 notation as a subscript has been dropped on each variable (y y-i-- etc.) with the understanding that the evaluated parameters are at the next time point. As the right hand side of each term in Eq. (10.25) indicates, the resulting equations are a coupled set of k equations in k unknowns which are the updated solutions of the differential equation variables at the next time increment. If the trapezoidal integration algorithm is selected instead of the backwards difference algorithm, the form of the equations is similar except for the following replacements ... 

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