2. 6. Numerical Integration 6.1 Definition of numerical integration. 6.2 Reasons to use numerical integration. 6.3 Formulas of numerical Integration. 6.4 Applications of numerical integration.

3. 6.1 Numerical Integration

4. Degree of Precision

5. 6.2 Reasons to use numerical integration Form of the function is not known. Integrand is not known clearly. Integrand is known but it is difficult or impossible to find antiderivative.

6. 6.3 Quadrature formula

7. Continued..

8. Trapezoidal Rule

9. Continued..

10. Simpson’s 1/3 rd Rule

11. Continued..

12. Simpson’s 3/8 th Rule

13. Continued..

14. Example A solid of revolution is formed by rotating about the x-axis, the lines x = 0 and x = 1.5, and a curve through the points with the following coordinates Estimate the volume of the solid formed, giving the answer to four decimal places by Trapezoidal rule, Simpson’s ⅓ and Simpson’s ⅜ rule of integration. x0.000.250.500.751.001.251.50 y1.00000.98960.95890.90890.84150.77350.7068

15. Solution X0.000.250.500.751.001.251.50 1.00000.97930.91950.82610.70810.59840.4996

16. Continued

17. Error Then the absolute relative approximate error obtained between the results from Trapezoidal rule and Simpson’s ⅓ rule is 0.18334 and the absolute relative approximate error obtained between the results from Simpson’s ⅓ rule and Simpson’s ⅜ rule is 0.04519

18. Comparison table Table:-Comparison of results of different methods of numerical Integration. Method of Integration Trapezoidal ruleSimpson’s ⅓ ruleSimpson’s ⅜ rule Volume3.75673.76363.7619 Absolute relative Approximate error ---------------0.183340.04519

19. Applications of numerical integration. Numerical integration is specially used in statistics in maximum likelihood & normal distribution. It is also used in cloud computing and to find out the definite integral when we are not knowing the form of the function etc.