How the number of points used influences the approximation

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Reference no: EM132249159

Math Assignment - Project:  Numerical Integration

Description: There are cases where a number for particular area under the curve is desired and hence a definite integral abf(x)dx needs to be solved; but, unfortunately, the Fundamental Theorem of Calculus cannot be used. In these cases numerical integration techniques can be used to get an approximate value that is good enough for a particular application.

"Good enough" means that the approximation has enough digits of precision, and that the error from the true value is relatively tiny.

Scope of Project: The goals of the project are to gain an understanding of (this is what you want to discuss in the write up):

  • How good a Numerical Integration technique is at approximating the true answer of a definite integral.
  • How the number of points used influences the approximation.
  • How the choice of method (shape) used influences the approximation.
  • How to evaluate the effectiveness of a method when the true solution cannot be obtained.
  • How spacing of the points influences the approximation.

Error:

We are interested in seeing how good numerical integration techniques are for calculating a definite integral; to do this we a notion of error.

Components:

1) Example with known antiderivative

For the function: f(x) = 4x3 - x4 - x2

Find the true value for the integral using Fundamental Theorem of Calculus 04f(x)dx.

Evaluate Riemann sums for n = 2, 4, 8, 16, 32. n corresponds to the number of rectangles used, or the number of slices taken, or the regions it is broken up into.

Evaluate Trapezoid rule for n = 2, 4, 8, 16, 32. n corresponds to the number of trapezoids used, or the number of slices taken, or the regions it is broken up into.

Create a table that compares the results for each method for each value of n, with the true solution found by using the definite integral.

2) Example with unknown antiderivative

This time a function with no antiderivative formula will be used. This function is the pdf for the standard normal curve (the bell curve in stats): f(x) = 1/√(2π) e-½x^2  

The goal is to approximate the definite integral 01.96 f(x)dx.

This is a real practical problem for which you (should) actually care about getting a good answer.

Evaluate Riemann sums for n = 2,4,8,16,32. n corresponds to the number of rectangles used, or the number of slices taken, or the regions it is broken up into.

Evaluate Trapezoid rule for n = 2,4,8,16,32. n corresponds to the number of trapezoids used, or the number of slices taken, or the regions it is broken up into.

Create a table that compares the results for each method for each value of n, with the "true solution" of 0.4750.

More digits 0.475002105; note this is the commonly accepted answer but it may or may not be super accurate.

3) Write up discussing the factors involved in finding a good approximation. Deductions for not spell checking etc.

4) Bonus: Simpson's rule, repeat the calculation using Simpson's rule (see text book).

5) Bonus: looking at how spacing of the intervals, that is using different widths for the subinterval affects the accuracy. For this let ?? = 8 and vary the widths of the sizes of the intervals. To receive credit you need to do 5 clearly different pattern of spacings for the subinterval widths keeping ?? fixed at ?? = 8 and do so for both of the Riemann sum and Trapezoid rule. The goal is to try to get each method as close to the true answer by varying the widths of the sub regions.

Be sure to give citation for ANY sources that you use (print, website, person).

Note - need to be three submissions, one hand in, one excel file to show calculations and one write up of a page and a half.

Attachment:- Assignment File.rar

Reference no: EM132249159

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Reviews

len2249159

3/6/2019 1:51:49 AM

There need to be three submissions, one hand in, one excel file to show calculations and one write up of a page an a half. A group of 2-3 people is preferred. Working by yourself is possible but definitely not recommended; if you intend to do this please contact me first. A group should turn in 1 copy of the following items: Paper copy of: write up, summary table of results any hand calculation work you might do. Electronic copy of: write up text document. Electronic copy of: Excel file calculations. (specifically, I want to see formulas in the sheet) Project is considered incomplete if any of those these are missing. A significant delay in providing any of these can result in reduced credit or a grade of zero for the project. Please: do not print out excel file.

len2249159

3/6/2019 1:51:41 AM

Scoring: (45pts) Look at accuracy in a case where an Antiderivative is available to compute the true solution: Calculate definite integral using Fundamental theorem of Calculus - 5pts, Evaluate Riemann sums for n=2,4,8,16,32 - 15pts, Evaluate approximation using trapezoid rule for n=2,4,8,16,32 - 15pts , Make a summary table that compares the error in the calculation 10pts. (40pts) Look at accuracy in a case where no Antiderivative is available: Evaluate Riemann sums for n=2,4,8,16,32 - 15pts, Evaluate approximation using trapezoid rule for n=2,4,8,16,32 - 15pts. Make a summary table that compares the error in the calculation - 10pts.

len2249159

3/6/2019 1:51:36 AM

(15pts) Write up discussing the factors involved in finding a good approximation - Text length should be one to one and half (1 to 1 1/2 pages) double spaced typed. Should consider both the shaped used AND number of regions, Explain what should be done when no Antiderivative is available but the area under the curve is needed. Again see scope of project from above to know what needs to be addressed. (Bonus 1) also use Simpson’s rule for approximating area under curve in parts 1 and 2. (Bonus 2) How the spacing of the sub intervals affects accuracy. (Bonus 3) 5pts Meeting before assignment due.

len2249159

3/6/2019 1:51:29 AM

Be sure to give citation for ANY sources that you use (print, website, person). The textbook, this handout and possibly office hours or the Math Lab should be enough to do the project. But outside sources are ok (as long as you cite them). Academic Conduct: See syllabus and Bentley code of conduct for more information. General Collaboration between members of different groups is discouraged. That means, DO NOT discuss the project with other groups from the class. DO NOT discuss the project with an upper classman, unless that upper classman is working in the Math Lab. If you have a question about the project or need clarification, you should probably be asking me about it first anyway.

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