Unformatted text preview: HOMEWORK – 4 Numerical Integration methods Show clean hand calculations steps, and do not use computers for the problems that require hand calculations 1. Consider the following integral (hand calculations, no EXCEL please): 0.5 (1 x ) 2 0.5 dx 0 a) Evaluate the exact integral using the analytical method (as you would do in a calculus class) and show that the true solution 0.478306. b) Evaluate the approximate value of the integral with n = 1, 2, 4, 8 intervals using the trapezoidal rule, and compare the true errors c) Reprocess the above results twice using the Richardson extrapolation formula for with h = 2, and (h/2) =1; and also h =8 and (h/2) =4. Compare the estimates of true errors. d) Evaluate the approximate value of the integral using the Simpson’s 1/3rd rule for n = 4 and estimate the true error. 2. Consider the following integral: 2 e x cos( x )dx , where x is in radians. 0 a) Evaluate the approximate value of the integral using the trap rule for n = 1, 2, 4, 8, 16. Use EXCEL spreadsheet. b) Evaluate the integral using the analytical method (calculus methods). c) Estimate the integral using the Simpson’s 1/3rd rule for n = 8, 16. Use EXCEL spreadsheet. 3. Solve problem‐1 using a VB code for trap rule (use method‐1, simple direct method discussed in ppt slides). Use delx = 0.05 or n=10 (part of lab work) 4. Stream cross‐sectional areas (A) are required for a number of tasks in water resources engineering, including flood forecasting and reservoir designing. Unless electronic sounding devices are available to obtain continuous profiles of the channel bottom, the engineer must rely on discrete depth measurements to compute A. An example of a typical stream cross section data is given below. Use two trapezoidal rule applications for (h=4 and h=2) and, Simpson’s 1/3rd rule to estimate the cross‐sectional area from this data. (Hand calculations) Distance (m) Depth (m) 0 2 4 6 8 10 12 14 16 18 20 0 1.8 2 4 4 6 4 3.6 3.4 2.8 0 1 5. A beam is subjected to a distributed loading as shown in the Figure 1. The length of the beam is about 8 m. Find the reaction, R and the moment, M at point x = 0. (Hand calculations, use Simpsons rule) Since, Fz 0 8
8 Reaction, R dw q ( x)dx 0
0
L
L [email protected] x0 dm q( x).xdx 0
0 Figure 1. Load distribution on the beam. 6. In surface water hydrology, hydrograph is a time record of stream flow rate at a watershed outlet. A hydrograph is a representation of how a watershed responds to a rainfall. The area under a hydrograph gives the total volume of water produced by the storm. Figure 2, shows a typical hydrograph. The flow observed at the outlet of a watershed at different times is given in the following table. Calculate the total volume of water produced by the storm in cubic feet. (Hand calculations, use trap rule). Be careful with your units!!! 4
5
6
7
8 9 10 11 12 13 14
Time(hr) 0 1 2 3 Q (cfs) 0 12 32 62 108 180 208 182 126 80 53 32 18 6 0 2 Hydrograph
250 Q (cfs) 200
150
100
50
0
0 2 4 6 8 10 12 14 Time (hrs) Figure 2. Hydrograph due to a storm 7. A rod subjected to axial load will be deformed, as shown in the stress‐strain curve in the Figure 3. The area under the curve from zero stress out to the point of rupture is called the Modulus of toughness of the material. It provides a measure of the energy per unit volume required to cause the material to rupture. As such, it is representative of the material’s ability to withstand an impact load. Use numerical integration to compute the modulus of toughness for the stress strain curve. (Hand calculations). Stress‐ strain curve
70
60 s, ksi 50
40
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20
10
0
0 0.05 0.1 0.15 0.2 0.25 e Figure 3. Stress‐strain curve for an axially loaded rod. e 0.02
0.05
0.10
0.15
0.20
0.25 2
8. The integral s, ksi 40.0
37.5
43.0
52.0
60.0
55.0 z e (t 2 )/2 dt is called the probability integral. For z=0.8, evaluate this 0 integral with the trapezoidal rule using 4 equal intervals. Use delz = 0.2, do hand calculations. 9. Use EXCEL to evaluate the following integral with the trapezoid rule with 16 intervals 3 0.002 ∗ 4 ∗ 0.1 sin 0.063 ∗ 10. An accelerating object (say your car) is traveling with variable velocity and the velocities of the object at various times are recorded in the following table. t, sec V, ft/sec 0 0 1 1.39 2 3.30 3 5.55 4 8.05 6 13.6 8 19.8 10 26.4 12 33.3 Determine the total distance travelled by the object after 12 seconds. Note: v(t) = dx/dt and hence the total distance travelled between t = 0 to t =12 can be calculated by integrating the velocity function. Use Simpson’s rule to solve the problem (Do hand calculations) 11. The following data were obtained for the variation of acceleration with time: t (sec) a (ft/sec2) 0.0
0.00
1.0
1.01
2.0
7.98
4.0
26.50
6.0
65.00
8.0
120.00
Given V(0) = 0, calculate the object velocity at t=8.0 seconds. Use trapezoidal rule (do hand calculations). Note: a(t) = dv/dt. You can integrate the function to get v(t). 12. Submit Lab problem #2 (polynomial integration lab problem) 13. Submit the results for the retaining wall lab problem (lab problem) 4 ...
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 Fall '09
 Clement
 Harshad number, hand calculations

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