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Unformatted text preview: Numerical Analysis in Engineering ME 140A, Fall 2007 Final Solution December 15, 2007 Problem 1 Use the trapezoidal rule to integrate I = Z 2 Z 3 xy dx dy (1) Use 3 intervals of equal size in the xdirection, and 2 intervals of equal size in the ydirection. Compare your result with the analytical result. Discuss the comparison. Solution: I = Z 2 y Z 3 x dx dy (2) First, use composite trapezoidal rule in xdirection: I = Z 2 y 0 + 1 × 1 + 1 × 2 + 1 2 × 3 dy = 9 2 Z 2 y dy (3) Now, use composite trapezoidal rule in ydirection: I = 9 2 0 + 1 × 1 + 1 2 × 2 = 9 (4) Next, we can find the integral analytically as below: I = Z 2 y Z 3 x dx dy (5) = Z 2 y x 2 2  3 dy (6) = 9 2 y 2 2  2 (7) = 9 (8) Comparing the results, we get exact agreement between the numerical and ana lytical result since the given function is linear both in terms of x and y. There fore, the trapezoidal rule can reproduce the functions correctly....
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This note was uploaded on 01/04/2011 for the course ME 140A taught by Professor Meiburg during the Spring '08 term at UCSB.
 Spring '08
 Meiburg

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