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Unformatted text preview: City University of Hong Kong Electronic Engineering Semester B 2007-2008 EE3108 Engineering Analysis Tutorial 7 The spectrum of a light bulb or the sun follows the following formula: f ( x) = x3 , e x −1 where x is a normalized frequency. The curve is plotted as below: 1.5 1 f(x) 0.5 0 0 1 2 3 4 5 x 6 7 8 9 10 The total amount of power is given by integrating the whole spectrum as: I =∫ ∞ x3 dx . ex −1 0 However, it is difficult to calculate the function when x is 0 or infinity. So an approximation is considered: I ≈∫ x3 dx . e x −1 1 9 The integration range is limited to 1 to 9 here, but the precision can be improved by using a wider range when necessary. (Note: we can show using the L’Hospital’s rule that f(x) = 0, but the computer will fail to recognize it.) EE 3108 Semster B 2007/2008 T7-1 S C Chan I. Trapezoidal rule Using the trapezoidal rule with 8 intervals, find I(x) by filling the following table: Step size h = ____________
x2 x1 x2 x1 ∫ f ( x)dx ≈ Thus, the total gives I = ______________ T7-2 II. Simpson’s rule Using the Simpson’s rule with 8 intervals (i.e. 4 pairs of intervals), find I(x) by filling the following table: Step size h = ____________
x2 x1 x2 x1 ∫ f ( x)dx ≈ Thus, the total gives I = ______________ What if the integration range is widened to 0.01 to 20.01 and the step size is decreased to 0.01? T7-3 ...
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- Spring '07
- Frequency, Kelsey Grammer, City University of Hong Kong Electronic Engineering Semester, I. Trapezoidal rule