Instructor: Hacker Name:Course: AME 302Chapter 9 Homework SetNumerical IntegrationInstructions: This homework is broke into three components: Part 1: true-false concept questions,Part 2: basic computational skills, and Part 3: derivation, analysis, and problem-solving questions.The rules for each component are listed below.Part 1: There is no partial credit given for true-false problems. Since these are concept questions, you donot need to show any work for these problems. There are 10 problems worth 40% (4 points each).Part 2: There is no partial credit given for multiple-choice problems. Although there is no partial crediton this assignment, you must show your work on all of the problems. If you fail to show workyou will receive a zero for the problem even if it is correct. There are 4 problems worth 40% (10points each).Part 3: On these problems you must show all of your work to receive any credit. If in doubt, write it out!Show your work as clearly as you can: if I can’t understand how you got an answer, I will notgive you credit for it. Remember, I know how to solve the problem; and to make matters worse, Ihave a lot of training in following logical arguments! There are 2 problems worth 20% (10 pointseach).Warning: The definition of “little or no work” will be determined by the instructor, not thestudent.Parts 1-3: Circle your answers here. Do not detach this sheet from the homework.1. T F2. T F3. T F4. T F5. T F6. T F7. T F8. T F9. T F10. T F11. a b c d e12. a b c d e13. a b c d e14. a b c d e15. a b c d e16. (10 pts)17. (10 pts)Attention: For all problems involving writing a MATLAB program you must turn inyour MATLAB code with the output to receive any credit!AME 302 chapter 9 hw set Copyright ©Wayne Hacker 2018. All rights reserved. 2Part 1: True-or-False Concept Questions ComponentNewton-Cotes Numerical Integration FormulasProblem 1. True or False: The composite trapezoidal rule uses linear functions toapproximate the underlying function over each subinterval.Problem 2. True or False: The composite trapezoidal rule cannot be applied to a dataset unless the underlying function that generated the data is known.Problem 3. True or False: Simpson’s 1/3 rule uses quadratic polynomials on eachsubinterval to approximate the underlying function, whereas Simpson’s 3/8 rule usescubic polynomials on each subinterval.Problem 4. True or False: For a small fixed step size h  1, Simpson’s 1/3 rule ismore accurate than the trapezoidal rule.Problem 5. True or False: For a small fixed step size h  1, Simpson’s 3/8 rule ismore accurate than Simpson’s 1/3 rule.Advanced Numerical Integration Formulas Involving FunctionsProblem 6. True or False: In order for Richardson Extrapolation to be applied to anumerical integration procedure, the error term in the procedure must have a predictableform in terms of the step size (or some other parameter).Problem 7. True or False: Romberg Integration uses Richardson Extrapolation toimprove the initial estimates that are made using the composite Simpson’s 1/3 rulemethod.Problem 8. True or False: The two-point Gaussian quadrature method,Z1−1f(x)dx ≈c0f(x0) + c1f(x1), uses a trapezoidal method where the two evaluation points x0and x1are the endpoints of the interval of integration.Problem 9. True or False: Gaussian quadrature is ideally suited for experimental databecause the underlying function that generated the data need not be known to apply themethod.Problem 10. True or False: Adaptive quadrature methods are ideally suited for work-ing with slow-varying functions.AME 302 chapter 9 hw set Copyright ©Wayne Hacker 2018. All rights reserved. 3Part 2: Basic Computational Skills ComponentNewton-Cotes Numerical Integration FormulasProblem 11. Consider the following elementary integral:I =Z40(1 − e−x) dx = (x + e−x)40= 3 + e−4≈ 3.018316 .Evaluate the integral using a the composite trapezoidal rule with n = 4 and compute thetrue percent relative error based on the exact solution to two decimal places.(a) t= 1.67% (b) t= 1.27%(c) t= 2.67% (d) t= 2.27%(e) None of theseProblem 12. Consider the following elementary integral:I =Z40(1 − e−x) dx = (x + e−x)40= 3 + e−4≈ 3.018316 .Evaluate the integral using Simpson’s 1/3 rule with n = 4 and compute the true percentrelative error based on the exact solution to two decimal places.(a) t= 0.16% (b) t= 0.26%(c) t= 0.36% (d) t= 0.46%(e) None of theseProblem 13. Consider the following elementary integral:I =Z40(1 − e−x) dx = (x + e−x)40= 3 + e−4≈ 3.018316 .Evaluate the integral using a single application of Simpson’s 3/8 rule and compute thetrue percent relative error based on the exact solution to two decimal places.(a) t= 0.9% (b) t= 1.1%(c) t= 1.3% (d) t= 1.5%(e) None of theseAME 302 chapter 9 hw set Copyright ©Wayne Hacker 2018. All rights reserved. 4Advanced Numerical Integration Formulas Involving FunctionsProblem 14. Consider the following elementary integral:I =Z40(1 − e−x) dx = (x + e−x)40= 3 + e−4≈ 3.018316 .Evaluate the integral using the two-point Gaussian quadrature approximation and com-pute the true percent relative error based on the exact solution to two decimal places.(a) t= 0.9% (b) t= 1.2%(c) t= 1.5% (d) t= 1.8%(e) None of theseProblem 15. Consider the following integral:I =Z300200z5 + ze−z/15dz .Use the Romberg integration algorithm to find R3,3rounded to four decimal places. Keepall of the digits until the end! For computational convenience, the first column of thetrapezoidal approximations are included to a more than sufficient level of accuracy in thetable below. That is, you will not need to compute beyond the level of accuracy of whatis given. MATLAB’s built-in integration function gives the value of the integral as I =1480.5685 to four place accuracy. What is the true absolute percent relative error basedon the exact solution to two significant figures?i O(h2i) O(h4i) O(h6i)1 348.00501404 X X2 1001.73124965 R2,2X3 1320.58477875 R3,2R3,3(a) t= 1.5% (b) t= 1.9%(c) t= 2.3% (d) t= 2.7%(e) None of theseAME 302 chapter 9 hw set Copyright ©Wayne Hacker 2018. All rights reserved. 5Part 3: Derivation, Analysis, and Problem-Solving ComponentProblem 16. The total mass of a variable density rod is given bym =ZL0ρ(x)Ac(x)dxwhere m = mass, ρ(x) = density, Ac(x) = cross-sectional area, and x = distance alongthe rod. The following data