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3. What is Integration? Integration The process of measuring the area under a curve. Where: f(x) is the integrand a= lower limit of integration b= upper limit of integration f(x) ab y x 3

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5. 5 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

6. 6 Cont. 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,

7. 7 Method Derived From Geometry The area under the curve is a trapezoid. The integral

8. 8 Example 1 The vertical distance covered by a rocket from t=8 to t=30 seconds is given by: Use single segment Trapezoidal rule to find the distance covered.

9. 9 Solution

10. 10 Solution (cont)

11. 11 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. 12 Cont. With Hence:

13. 13 Cont. 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:

14. 14 Multiple Segment Trapezoidal Rule The integral I can be broken into h integrals as: Applying Trapezoidal rule on each segment gives:

15. 15 Example 2 The vertical distance covered by a rocket from to seconds is given by: Use two-segment Trapezoidal rule to find the distance covered.

16. 16 Solution a) The solution using 2-segment Trapezoidal rule is

17. 17 Solution (cont) Then:

18. 18 Solution (cont) 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

19. 19 Example 3 Use Multiple Segment Trapezoidal Rule to find the area under the curve from to. Using two segments, we get and

20. 20 Solution Then:

21. 21 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:

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