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 _a∫^bf(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 ₀∫⁴f(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 ₀∫^1.96 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