1. 1 Trapezoidal Rule of Integration

2. http://numericalmethods.eng.usf.edu2 What is Integration Integration: The process of measuring the area under a function plotted on a graph. Where: f(x) is the integrand a= lower limit of integration b= upper limit of integration

3. http://numericalmethods.eng.usf.edu3 Basis of Trapezoidal Rule Trapezoidal Rule is based on the Newton-Cotes Formula that states if one can approximate the integrand as an n th order polynomial… where and

4. http://numericalmethods.eng.usf.edu4 Basis of Trapezoidal Rule Then the integral of that function is approximated by the integral of that n th order polynomial. Trapezoidal Rule assumes n=1, that is, the area under the linear polynomial,

5. http://numericalmethods.eng.usf.edu5 Derivation of the Trapezoidal Rule

6. http://numericalmethods.eng.usf.edu6 Method Derived From Geometry The area under the curve is a trapezoid. The integral

7. http://numericalmethods.eng.usf.edu7 Example 1 The vertical distance covered by a rocket from t=8 to t=30 seconds is given by: a)Use single segment Trapezoidal rule to find the distance covered. b)Find the true error, for part (a). c)Find the absolute relative true error, for part (a).

8. http://numericalmethods.eng.usf.edu8 Solution a)

9. http://numericalmethods.eng.usf.edu9 Solution (cont) a) b) The exact value of the above integral is

10. http://numericalmethods.eng.usf.edu10 Solution (cont) b) c) The absolute relative true error,, would be

11. http://numericalmethods.eng.usf.edu11 Multiple Segment Trapezoidal Rule In Example 1, the true error using single segment trapezoidal rule was large. We can divide the interval [8,30] into [8,19] and [19,30] intervals and apply Trapezoidal rule over each segment.

12. http://numericalmethods.eng.usf.edu12 Multiple Segment Trapezoidal Rule With Hence:

13. http://numericalmethods.eng.usf.edu13 Multiple Segment Trapezoidal Rule The true error is: The true error now is reduced from -807 m to -205 m. Extending this procedure to divide the interval into equal segments to apply the Trapezoidal rule; the sum of the results obtained for each segment is the approximate value of the integral.

14. http://numericalmethods.eng.usf.edu14 Multiple Segment Trapezoidal Rule Figure 4: Multiple (n=4) Segment Trapezoidal Rule Divide into equal segments as shown in Figure 4. Then the width of each segment is: The integral I is:

15. http://numericalmethods.eng.usf.edu15 Multiple Segment Trapezoidal Rule The integral I can be broken into h integrals as: Applying Trapezoidal rule on each segment gives:

16. http://numericalmethods.eng.usf.edu16 Example 2 The vertical distance covered by a rocket from to seconds is given by: a) Use two-segment Trapezoidal rule to find the distance covered. b) Find the true error, for part (a). c) Find the absolute relative true error, for part (a).

17. http://numericalmethods.eng.usf.edu17 Solution a) The solution using 2-segment Trapezoidal rule is

18. http://numericalmethods.eng.usf.edu18 Solution (cont) Then:

19. http://numericalmethods.eng.usf.edu19 Solution (cont) b) The exact value of the above integral is so the true error is

20. http://numericalmethods.eng.usf.edu20 Solution (cont) c) The absolute relative true error,, would be

21. http://numericalmethods.eng.usf.edu21 Solution (cont) Table 1 gives the values obtained using multiple segment Trapezoidal rule for: nValueEtEt 111868-8077.296--- 211266-2051.8535.343 311153-91.40.82651.019 411113-51.50.46550.3594 511094-33.00.29810.1669 611084-22.90.20700.09082 711078-16.80.15210.05482 811074-12.90.11650.03560 Table 1: Multiple Segment Trapezoidal Rule Values

22. http://numericalmethods.eng.usf.edu22 Example 3 Use Multiple Segment Trapezoidal Rule to find the area under the curve from to. Using two segments, we get and

23. http://numericalmethods.eng.usf.edu23 Solution Then:

24. http://numericalmethods.eng.usf.edu24 Solution (cont) So what is the true value of this integral? Making the absolute relative true error:

25. http://numericalmethods.eng.usf.edu25 Solution (cont) nApproximate Value 10.681245.9199.724% 250.535196.0579.505% 4170.6175.97830.812% 8227.0419.5467.927% 16241.704.8871.982% 32245.371.2220.495% 64246.280.3050.124% Table 2: Values obtained using Multiple Segment Trapezoidal Rule for: