Simpson’s 3/8 Rule

In applying Simpson s rule, over the interval [a, i>] of the independent variable, the interval is partitioned into an even number of subintervals and three consecutive points are used to determine the unique parabola that covers the area of the first... 

Figure 1-3 Areas Under a Parabolic Arc Covering Two Subintervals of a Simpson s Rule Integration.

The area under a parabolic arc concave upward is bh, where b is the base of the figure and h is its height. The area of a parabolic arc concave downward is jh/t. The areas of parts of the figure diagrammed for Simpson s rule integration are shown in Fig. 1-3. The area A under the parabolic arc in Fig. 1-3 is given by the sum of four terms ... 

Our Simpson s rule program is written in QBASIC (Appendix A). Today s computer world is full of complicated and expensive software, some of which we shall use in later chapters. Unfortunately, it is not hard to find software that is overpriced and overwritten (which we shall not use). Although it is not appropriate to recommend software in a book of this kind, the simple software used here has been used for several years in both a teaching and a research setting. It works. 

PRINT " Simpson s Rule integration of the area under y = f (x) " DEF fna (x) = 100 - X 2 DEF fna lets you put any function you like here. PRINT "input limits a, andb, and the number of iterations desired n"... 

The integral of the Gaussian distribution function does not exist in closed form over an arbitrary interval, but it is a simple matter to calculate the value of p(z) for any value of z, hence numerical integration is appropriate. Like the test function, f x) = 100 — x, the accepted value (Young, 1962) of the definite integral (1-23) is approached rapidly by Simpson s rule. We have obtained four-place accuracy or better at millisecond run time. For many applications in applied probability and statistics, four significant figures are more than can be supported by the data. 

This is not E, i , of course you must find the minimum energy by systematic variation of ot. Alternatively, a QBASIC or TBASIC program can be written to integrate Eq. (6-28) by Simpson s rule. 

This definite integral can be evaluated numerically by the use of Simpson s rule to obtain hr= 0.305 m (1 ft). 

Equation 4.44 is solved using Simpson s Rule to integrate the area of overlap. The method is easily transferred to eomputer eode for high aeeuraey. See Appendix XII for a diseussion of Simpson s Rule used for the numerieal integration of a funetion. Equation 4.44 permits the ealeulation of reliability for any eombinations of distributions for stress and strength provided the partial areas of G and H ean be found. 

Substituting in the parameters for both stress and strength into equation 4.52 and solving using Simpson s Rule (integrating between the limits of 1 and 1000, for example) gives that the reliability is ... 

Next, solving equation 4.35 direetly using Simpson s Rule for as deseribed by Freudenthal et al. (1966) ... 

The limits of integration are from the expected minimum value of yield strength, xos = 272.4 MPa to 1000 MPa, representing oo. The solution of this equation numerically using Simpson s Rule is described in Appendix XII. For the case when d = 20 mm and the number of load applications n = 1000, the reliability, 7 , is found to be ... 

One commonly used technique which gives high accuracy is Simpson s Rule (more correctly called Simpson s Rule). Here the area under the curve is divided into equal segments of width, h. For an even number of segments, m, we can divide the range of interest (MAX — MIN) into ordinates xq to where the number of ordinates is odd. 

Figure 2 Graphical representation of Simpson s Rule using multiple segments...

Using Simpson s Rule outlined above, the maximum limit, oo, is dilfieult to work with and an appropriate value refleeting the problem should replaee it. For argument s sake, we will give it a value of 700 MPa. Therefore, the reliability ean be determined given that ... 

Now applying Simpson s Rule, let s start with a relatively low number of segments, m =10 ... 

Area under a Function Calculated using Simpson s Rule (written in Visual Basic)... 

The above integral can be evaluated either analytically or numerically by applying Simpson s rule. The following provides a summary of the major working equations for compressor analysis ... 

The area of l/(-r ) versus or plots is determined for all design equations for bateh, CFSTR, and plug flow reaetors by employing the Simpson s rule. 

Plot curve of 1/ (yj - Xj) versus xj see Figure 8-38, graphical integration by Simpson s rule. 

Figure 8-38. Graphical integration of Rayleigh or similar equation by Simpson s Rule, for Example 8-14.

If the function f(x) is approximated by parabolas, Simpson s Rule is obtained, by which (the number of panels n being even)... 

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.

---