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Mid1Solution

Mid1Solution - Numerical Analysis in Engineering ME 140A...

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Numerical Analysis in Engineering ME 140A, Fall 2008 Midterm #1 Solution By: Mohamad M. Nasr-Azadani [email protected] December 3, 2008 Problem 1 a) Since h = h ( r ) (only a function of r ), volume of the given shape can be found as V = Z Z A h ( r ) dA (1) V = Z R 0 h ( r )2 πrdr (2) Note that “ dV = h ( r )2 πrdr ” is the differential volume of a cylinder with radius r , thickness dr and height h ( r ) (see figure 1). b) Total mass of the pollutant in the reservoir can be found by M = Z Z Z V c ( r ) dV (3) M = Z R 0 c ( r ) h ( r )2 πrdr (4) Note that in equation (4), “ dM = c ( r ) h ( r )2 πrdr ” is the differential mass of the pollutant in the differential volume (see figure 1). 1
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Figure 1: Left: Side view of the reservoir. Right: Three dimensional view of the element. Problem 2 a) Analytical integral for the given function f ( x ) = cos 2 πx is found as Z 1 0 (cos 2 πx ) dx = sin 2 πx 2 π fl fl fl fl 1 0 = 0 . (5) b) Using the Trapezoidal rule for n = 1, the given integral is found as Z 1 0 (cos 2 πx ) dx = 1 - 0 2 [ f (0) + f (1)] = 1 . (6) c) Using the Trapezoidal rule for n
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