EGM_3344_Exam4_Solutions_2009

EGM_3344_Exam4_Solutions_2009 - Sorting Number: _ Problem 3...

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Sorting Number: ______________________ Fall 2009 1 Problem 3 (Handwritten) – Numerical Integration of Functions. (13 points) Both parts of this problem involve numerical evaluation of the following integral: 2 1.5 1 (2 ) x I xe d x =+ The integrand of this problem is plotted below over the interval [-1, 3]: -1 0 1 2 3 0 2 4 6 8 x f(x) a) The table below provides values of the integrand at the points specified in the plot above: x f ( x ) -1 8.0 -0.5 3.7 0 2.0 1 1.4 2 2.1 3 3.0 Using all of the points provided in this table, solve this integral using Simpson's 1/3 rule as many times as possible and the trapezoidal rule elsewhere. Record your final integral result in the space provided below: I = _____________________________ Part b) on next page.
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Sorting Number: ______________________ Fall 2009 2 b) The table below provides numerical integration results for this problem obtained by using the trapezoidal rule with step sizes of 1, 2, and 4 across the entire interval [-1, 3]: Number of Segments Step size h Trapezoid Result 1 4 22 2 2 13.8 4 1 11 Using the initial trapezoid results from the table above, re-solve this integral by implementing Romberg integration as many times as necessary to obtain the highest accuracy integral possible. Record your Romberg results in the table above. Solution: a) Apply Simpson’s 1/3 rule to points 1, 2, 3 with step size 0.5 and to points 3, 4, 5 (or 4, 5, 6) with step size 1. Then apply Trapezoidal rule to points 5 and 6 (or 3 and 4) to complete the integral. First Simpson = (0.5/3)*(8.0+4*3.7+2.0) = 4.1333 Second Simpson = (1/3)*(2.0+4*1.4+2.1) = 3.2333 Trapezoid = (1/2)*(2.1+3.0) = 2.5500 Total = 9.9167 or First Simpson = (0.5/3)*(8.0+4*3.7+2.0) = 4.1333 Trapezoid = (1/2)*(2.0+1.4) = 1.7000 Second Simpson = (1/3)*(1.4 +4*2.1+3.0) = 4.2667 Total = 10.1000 b) Let I4 be the trapezoid results using a step size of 4, I2 the trapezoid results using a step size of 2, and I1 the trapezoid results using a step size of 1.
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EGM_3344_Exam4_Solutions_2009 - Sorting Number: _ Problem 3...

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