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Homework 16 Solution

# Homework 16 Solution - Lecture 16 MEEN 357 Homework...

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Lecture 16 MEEN 357 Homework Solution 1 Lecture-16-Homework-Solution-2009.doc RMB 1. Problem 17.3 of the textbook Solution: This problem is essentially the same as Problem 17.2 that you worked on the homework for Lecture 15. You are asked to evaluate the integral () 2 0 63 c o s I xd x π =+ (16.1) a) Analytically b) Single application of trapezoidal rule. c) Composite trapezoidal rule with 2 N = and 4 N = . d) Single application of Simpson’s 1/3 rule. e) Composite Simpson’s 1/3 rule with 4 N = . f) Simpson’s 3/8 rule. The book does not specify a single application of the 3/8 rule. For simplicity, we shall work the single application case. You are asked to determine the percent relative error for parts b) through f) based upon the answer to part a). Part a) Exact The integral of c o s f xx is elementary and is given by ( ) () c o s 6 3 s i n x dx x x C += + + (16.2) Therefore, 2 2 0 0 c o s s i n 6 3 s i n 3 3 22 12.4248 x a x Ix d x x x ππ = = ⎛⎞ = + = + = + ⎜⎟ ⎝⎠ = (16.3) Part b) Single application of trapezoidal rule The formula for the approximate integral is ( ) ( ) 2 b f af b Ib a + =− (16.4) Since ( ) ( ) 09 6 2 fa f fb f = = = = (16.5)

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Lecture 16 MEEN 357 Homework Solution 2 Lecture-16-Homework-Solution-2009.doc RMB it follows from (16.4) that () ( ) ( ) 96 1 5 11.78 22 2 4 b fa fb Ib a π + ⎛⎞ + =− = = = ⎜⎟ ⎝⎠ (16.6) and true error is 12.4248 11.78 100 100 5.18% 12.4248 ab t a II I ε == = (16.7) Part c) Composite trapezoidal rule with 2 N = and 4 N = .
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Homework 16 Solution - Lecture 16 MEEN 357 Homework...

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