Reference no: EM132351902
Numerical and Algebraic integration
You have recently learnt about the numerical approximation technique called the trapezoidal rule. You will have also learnt about algebraic integration for a range of functions.
Your task is to investigate the difference between algebraic integration and numerical techniques. You will consider both the trapezoidal ruleand one of the numerical techniques from the suggested list below.
• Simpson's rule
• The Monte Carlo method
• Any other method you have found (discuss with your teacher first)
Task 1 (Investigate your chosen numerical technique)
Describe the rule that you have chosen from the options above, including a simple example to demonstrate how it works. Include consideration of:
• Development of the generalisation of the given rule including a full discussion of varying the rules parameters.
• State the limitations of the rule that you have chosen.
• Justify how it works.
• Verify your chosen example using another technique.
Task 2 (Comparing your chosen rule and the trapezoidal rule with algebraic methods)
Compare, throughat least three (3) different functions, the precision of your results obtained, using your chosen rule, the trapezoidal rule and algebraic integration. Where possible verify your statements about accuracy through generalisations.
Consider demonstrating;
• In which situations each method is the most appropriate. Include examples of when numerical rules and algebraic integration are most suitable.
• Under what conditions accuracy can be improved.
• Justify the reasonableness of your conclusions.
• Outline the strengths and weaknesses of each method.
Often numerical approximation techniques will cause an over estimation or underestimation. Consider a variety of functions,both increasing and decreasing as well as any relevantresults from calculus.
Investigate when the trapezoidal rule, the midpoint rule and your chosen rule will cause an under or over estimation. Use diagrams to explain why this is the case and algebraically justify your conclusions.
Task 3
Using a suitable map and chosen scale find the area of acountry or region. State, province, territory et).Use your results to verify the stated areaof this country given by a reputable source. Outline the reasonableness of your result and any assumptions that you have made.
Attachment:- Numerical Integration Techniques.rar