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Chapter 7 – Techniques of Integration

Chapter 7 – Techniques of Integration. 7.7 Approximation Integration. Why do we use Approximate Integration?. There are two situations in which it is impossible to find the exact value of a definite integral : When finding the antiderivative of a function is difficult or impossible

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Chapter 7 – Techniques of Integration

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  1. Chapter 7 – Techniques of Integration 7.7 Approximation Integration 7.7 Approximation Integration

  2. Why do we use Approximate Integration? • There are two situations in which it is impossible to find the exact value of a definite integral: • When finding the antiderivative of a function is difficult or impossible • If the function is determined from a scientific experiment through instrument readings or collected data. There may not be a formula for the function. 7.7 Approximation Integration

  3. Approximate Integration For those cases we will use Approximate Integration • We have used approximate integration on chapter 5 when we learned how to find areas under the curveusing the Riemann Sums. We used Left, Right and Midpoint rules. • Now we are going to learn two new methods: The Trapezoid Rule and Simpson’s Rule • Let’s compare the approximation methods. 7.7 Approximation Integration

  4. Midpoint Rule • Remember, the midpoint rule states that where and 7.7 Approximation Integration

  5. Trapezoidal Rule • The Trapezoid rule approximates the integral by averaging the approximations obtained by using the Left and Right Endpoint Rules: where and 7.7 Approximation Integration

  6. Example 1 • Use (a) the Midpoint Rule and (b) the Trapezoidal Rule with n = 5 to approximate the integral below. Round your answer to six decimal places. 7.7 Approximation Integration

  7. Error Bounds in MP and Trap Rules • Suppose for a ≤ x ≤ b. If ET and EM are the errors in the Trapezoidal and Midpoint Rules, then 7.7 Approximation Integration

  8. Example 2 • Find the error in the previous problem. • Previous problem: Use (a) the Midpoint Rule and (b) the Trapezoidal Rule with n = 5 to approximate the integral 7.7 Approximation Integration

  9. Example 3 • How large should we take n in order to guarantee that the Trapezoidal Rule and Midpoint Rule approximations are accurate to within 0.0001 for the integral below? 7.7 Approximation Integration

  10. Simpson’s Rule • Simpson’s Rule uses parabolas to approximate integration instead of straight line segments. where and n is even. 7.7 Approximation Integration

  11. Error Bounds in Simpson’s Rule • Suppose for a ≤ x ≤ b. If ES is the error involved using Simpson’s Rule, then 7.7 Approximation Integration

  12. Example 4 • Use the (a) Midpoint Rule and (b) Simpson’s Rule to approximate the given integral with the specified value of n. Round your answers to six decimal places. Compare your results to the actual value to determine the error in each approximation. 7.7 Approximation Integration

  13. Example 5 • Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of n. Round your answers to six decimal places. 7.7 Approximation Integration

  14. Example 6 • Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of n. Round your answers to six decimal places. 7.7 Approximation Integration

  15. Example 7 – pg. 516 # 16 • Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of n. Round your answers to six decimal places. 7.7 Approximation Integration

  16. Example 8 – pg. 516 # 20 • (a) Find the approximations T10 and M10 for the above integral. • (b) Estimate the errors in approximation of part (a). • (c) How large do we have to choose n so that the approximations Tn and Mn to the integral part (a) are accurate to within 0.0001? 7.7 Approximation Integration

  17. Example 9 – pg. 518 # 37 • The table (supplied by San Diego Gas and Electric) gives the power consumption P in megawatts in San Diego County from midnight to 6:00 AM on December 8, 1999. Use Simpson’s Rule to estimate the energy used during that time period. (Use the fact that power is the derivative of energy.) 7.7 Approximation Integration

  18. Book Resources • Video Examples • Example 2 – pg. 510 • Example 3 – pg. 511 • Example 5 – pg. 513 • More Videos • Using the Midpoint Rule to approximate definite integrals, Part I • Using the Midpoint Rule to approximate definite integrals, Part 2 • Using the Midpoint Rule to approximate definite integrals, Part 3 • The Trapezoidal Rule • Using the Trapezoidal Rule to approximate an integral • Errors in the Trapezoidal Rule and Simpson’s Rule • Simpson’s Rule • Using Simpson’s Rule to Approximate an Integral 7.7 Approximation Integration

  19. Book Resources • Wolfram Demonstrations • Comparing Basic Numerical Integration Methods 7.7 Approximation Integration

  20. Web Links • http://youtu.be/JGeCLfLaKMw • http://youtu.be/z_AdoS-ab2w • http://www4.ncsu.edu/~acherto/NCSU/MA241/sec59.pdf • http://youtu.be/zUEuKrxgHws 7.7 Approximation Integration

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