Integral trapezoidal approximation

The integrals in equations (C) and (F) can also be evaluated numerically using the E-Z Solve software (file exl5-7.msp). The calculated value of t is 15.8 s, slightly less than that obtained using the trapezoidal approximation. 

The equation is solved at discrete points x , and the integral is approximated by the trapezoidal rule to give the following system of equations ... 

As expected, the trapezoidal approximation gives more nearly correct values than does the bar-graph approximation, for the same number of panels. For 10 panels, the trapezoidal approximation gives a result of 0.135810 for the integral in Eq. (5.54). For 100 panels, the trapezoidal approximation is correct to five significant digits. 

Note that this formula provides quite good approximation if the function to be integrated is smoothly enough and the number of intervals is enough large as well. However, the error in the trapezoid approximation accoimt for the fact that the interpolation is based on a single interval (so with n= I the above definition) and therefore reads for a single interval as ... 

Example 7.16. Using the trapezoidal approximation with five panels, calculate the value of the integral... 

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. 

The approximation of the integral on the right-hand side of (5) can be done using various quadrature formulae, for instance, by the formula of trapezoids... 

Equations (A), (B) and (C) are used in the algorithm to obtain the information required. Step (3) is used to calculate kA from equation (B), and step (4) is not required. Results are summarized in Table 12.2, for the arbitrary step-size in fA indicated G = tl[kA 1 -/A)j, and G represents the average of two consecutive values of G. The last column lists the time required to achieve the corresponding conversion in the second column. These times were obtained as approximations for the value of the integral in equation (A) by means of the trapezoidal rule ... 

Evaluate the integral in equation 21.6-5 by means of the E-Z Solve software or an approximation such as the trapezoidal rule, as in Example 21-5. 

The basic strategy to accomplish this goal is to develop discrete approximations for integration (like rectangular or trapezoidal) in terms of functions of z. These are called z forms, and they are different than z transforms. Then we substitute the z form for 1/s in G(j,. So the first thing we must do is develop discrete approximations for integration (1/s). 

TABLE 3.1 Approximations to the Integral J j sinf-n-x) dx = 1 Using the Trapezoidal and Legendre Quadrature Methods... 

This formula, termed the RL-algorithm, based as it is on a connect-the-dots approximation, is the semi-integration equivalent of the trapezoidal formula of integration. 

Approximating the integral on the right-hand side of Eq. (B.6) by a closed quadrature formula89 such as the trapezoidal rule, one obtains. 

In order to obtain the degree of cure and rate of curing, we must first measure the reaction. This is typically done using a differential scanning calorimeter (DSC) as explained in Chapter 2. Typically, several dynamic tests are performed, where the temperature is increased at a constant rate and heat release rate (Q) is measured until the conversion is finished. To obtain Qt we must calculate the area under the curve Q versus t. Figure 7.17 shows four dynamic tests for a liquid silicone rubber at heating rates of 10, 5, 2.5 and 1 K/min. The trapezoidal rule was used to integrate the four curves. As expected, the total heat Qt is the same (more or less) for all four tests. This is to be expected, since each curve was represented with approximately 400 data points. 

Numerical integration of a variable / measured at a set of equally spaced values of the independent variable x. The integral 7(1,7) = [ydx is approximated with both the trapezoidal rule (a) and Simpson s one-third rule b). In each case, the value of Y is given by the area under the heavy lines. The light lines in b) represent extensions of the three parabolic sections that are used to construct this approximation. 

In cases where n is divisible neither by 2 nor by 3, the range of integration may be split into two parts, one for Simpson s one-third rule, the other for Simpson s three-eighths rule. Alternatively, if the curve is approximately linear in one or two intervals, the trapezoidal rule may be used in these intervals. 

Yeh KC, Kwan KC. A comparison of numerical integrating algorithms by trapezoidal, Lagrange, and spline approximation. J Pharmacokinet Biopharm 1978 6 79-98. 

The trapezoidal rule is an approximation, as are all quadrature formulas. Figure A.3-1 illustrates the nature of the error introduced by its use. The integral to be evaluated... 

Trapezoidal rule (two-point) (Figure A-2). This method is one of the simplest and most approximate, as it uses the integrand evaluated at the limits of integration to evaluate the integral ... 

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. 

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