1. CSE 330 : Numerical Methods Lecture 15: Numerical Integration - Trapezoidal Rule Dr. S. M. Lutful Kabir Visiting Professor, BRAC University & Professor (on leave) IICT, BUET

2. What is Integration? Integration: Prof. S. M. Lutful Kabir, BRAC University 2 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. 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… Prof. S. M. Lutful Kabir, BRAC University 3 where and 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,

4. Method Derived From Geometry Prof. S. M. Lutful Kabir, BRAC University 4 The area under the curve is a trapezoid. The integral

5. Example 1 The vertical distance covered by a rocket from t=8 to t=30 seconds is given by: Prof. S. M. Lutful Kabir, BRAC University 5 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).

6. Integration of Velocity of Rocket Prof. S. M. Lutful Kabir, BRAC University 6 a)

7. Solution (cont) Prof. S. M. Lutful Kabir, BRAC University 7 a) b) The exact value of the above integral is

8. Solution (cont) Prof. S. M. Lutful Kabir, BRAC University 8 b) c) The absolute relative true error,, would be

9. Multiple Segment Trapezoidal Rule Prof. S. M. Lutful Kabir, BRAC University 9 In previous example, 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.

10. Multiple Segment Trapezoidal Rule Prof. S. M. Lutful Kabir, BRAC University 10 With Hence: The true error now is reduced from -807 m to -205 m.

11. Multiple Segment Trapezoidal Rule Prof. S. M. Lutful Kabir, BRAC University 11 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:

12. Multiple Segment Trapezoidal Rule Prof. S. M. Lutful Kabir, BRAC University 12 The integral I can be broken into n integrals as: Applying Trapezoidal rule on each segment gives:

13. Example 2 Prof. S. M. Lutful Kabir, BRAC University 13 The vertical distance covered by a rocket from t=8 to t=30 seconds is given by: a) Use n-segment Trapezoidal rule to find the distance covered (n=1 to 8). b) Find the true error, for part (a). c) Find the absolute relative true error, for part (a).

14. Solution Prof. S. M. Lutful Kabir, BRAC University 14 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

15. Thanks 15 Prof. S. M. Lutful Kabir, BRAC University