The Trapezoid

Now we take the rectangular wave discussed above and make it a little more realistic by introducing nonzero rise and fall times. By a similar procedure as for a rectangular wave, we can get the following equation (for the case of equal rise and fall times)  

Clearly we will get two break points now. The first break point occurs at 

Since n = frequency of harmonic/fundamental frequency, that is, n = f x T, we get the corresponding break frequency to be 

Note that n must be an integer to have any physical meaning. The first breakpoint, therefore, may not be even visibly apparent. What we will perceive is that the envelope ramps down almost from the lowest frequency, at the rate of 20 dB/decade. 

We can ask — when does ni become higher than n = 2 We can solve to get the condition 

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This method approximates/(x) by a parabola on each subintei val. This rule is generally more accurate than the trapezoidal rule. It is the most widely used integration formula. 

Example Evaluate the same integral (3-75) using the trapezoid rule and then apply the Romberg method. To achieve four-digit accuracy, any result from J9 through Li are suitable, even though the base results (Z1-Z4) are not that close. 

Errors are proportional to At for small At. The trapezoid rule is a second-order method. 

Errors are proportional to At for small At. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. 

This capacitor experiences the same current waveform at the power switch, which is a trapezoid with an initial current of about 1A rising to 2.8 A with very sharp edges. This capacitor has much more rigorous operating conditions than the output filter capacitor. I will estimate the RMS value of the trapezoidal current waveform as a piecewise superposition of two waveforms, a rectangular 1A peak waveform and a triangular waveform with a 1.8 A peak. This yields an estimated RMS value of 1.1 A. The value of the capacitor is then calculated as ... 

A feed and wash boxes that may be set in a slight slope so that the distribution over the trapezoidal surface will be proportional to the pan s area. This will ensure that the formed cake is even in thickness and that the applied wash liquids are distributed evenly. 

The diserete data ean be analyzed from the residenee time distribution by using either the histogram method or the trapezoidal rule... 

In the trapezoidal rule method, the area under the response eurve is ... 

Table 8-4 shows tlie details of tlie suimuation, whieh is perfonued using the trapezoidal rule. The internal age distribution 1(6) is obtained from... 

The original Simpson s formula without end correction may be generalized in a similar way as the trapezoidal formula for n = 2 panels, using Ax = (b - a)/2 and increasing k until sufficient accuracy is achieved, where... 

Then the trapezoidal rule applied to the interval from 0 to 1 corresponds to... 

Two simple numerical methods are often used to determine the area under the curve that equals the desired integral. They involve the use of the trapezoidal rule and Simpson s rule. 

Figure A1.5 Graphical integrations using (a) the trapezoidal rule and (b) Simpson s rule.

Another approach is to use Simpson s rule instead of the trapezoidal rule for the integration. With Simpson s rule, three successive points, e.g,, n), are... 

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. 

For / equally sized increments in the radial direction, the general form for the trapezoidal rule is... 

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. 

The following Excel macro illustrates the use of the trapezoidal rule for evaluating both the numerator and denominator in this equation. 

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. 

Similarly we can estimate the specific secretion rate. It is obvious from the previous analysis that an accurate estimation of the average specific rates can only be done if the integral Jxvdt is estimated accurately. If measurements of biomass or cell concentrations have been taken very frequently, simple use of the trapezoid rule for the computation of Jxvdt may suffice. If however the measurements are very noisy or they have been infrequently collected, the data must be first smoothed through polynomial fitting and then the integrals can be obtained analytically using the fitted polynomial. 

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. 

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