CH-05 (Pao - Engineering Analysis), страница 4
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In Figure 8, the shape and dimensions of a pyramid are described by thecoordinates of the five points (Xi,Yi) for I = 1,2,…,5. For application ofnumerical integration to determine its volume by either trapezoidal orSimpson’s rule, we have to partition the projected plane P2P3P4P5 into agridwork. At each interception point of the gridwork, (X,Y), the heightZ(X,Y) needs to be calculated which requires knowing the equationsdescribing the planes P1P2P3, P1P3P4, P1P4P5and P1P5P2.
The equation ofa plane can be written in the form of 2(X–a) + m(Y–b) + n(Z–c) = 1where (a,b,c) is a point on the plane and (2,m,n) are the directional cosines© 2001 by CRC Press LLCFIGURE 8. Problem 6.7.8.9.10.of the unit normal vector of the plane.3 Apply the equation of plane andassign proper values for the coordinates (Xi,Yi) describing the pyramid,and then proceed to write a FUNCTION Z(X,Y) to determine its volumeby using program Volume.Find the volume under the surface z = 3x2–4y + 15 over the base area of0≤x≤2 and 1≤y≤2 by applying Simpson’s Rule along the x-direction usingan increment of x = 1, and Trapezoidal Rule along the y-direction usingan increment of y = 0.25.How do you find the volume under the plane z = 2x–0.5y and above therectangular area bounded by x = 0, x = 1, y = 0, and y = 2 numericallyand not by actually integrating the z function? Explain which method andstepsizes in x and y directions you will use, give the numerical result anddiscuss how accurate it is.Use the function FuncZnew which defines the equation Z = 2X + 3Y2 +4 and plot the Z surface for 0≤X≤2 and 1≤Y≤2 by applying mesh ofMATLAB.
Experiment with different increments of X and Y.Modify the use of mesh by defining a vector {S} = [SX SY SZ} containingthe values of scaling factors for the three coordinate axes and then entermesh(Z,S) to try to improve the appearance of a hemisphere, better thanthe one shown in Figure 2. Referring to Figure 2, the lowest point is theoriginal and the X-axis is directed to the right (width), Y-axis is directedto the left (depth), and Z-axis is pointing upward (height). Since thehemisphere has a radius equal to 2 and by actually measuring the width,depth, and height to be in the approximate ratios of 2 7/8”: 2 7/16”: 23/4”.
Based on these values, slowly adjust the values for SX, SY and SZ.© 2001 by CRC Press LLCFIGURE 9. Problem 11.11. Figure 9 is obtained by using mesh to plot the surface Z = 1.5Re–2R andR = (X2 + Y2)H for –15≤X,Y≤15 with increment of 1 in both X and Ydirections. Try to generate this surface by interactively entering MATLABcommands. Apply the m file volume and modify the function FuncZnewto accommodate this new integrand function to calculate the volume ofthis surface above the 30x30 base area.12. Apply Mathematica to solve Problem 6.13. Apply Mathematica to solve Problem 7.14. Apply Mathematica to solve Problem 9.15. Apply Mathematica to solve Problem 11.5.5 REFERENCES1.
M. Abramowitz and I. A. Stegum, editors, Handbook of Mathematical Functions withFormulas, Graphs and Mathematical Tables, National Bureau of Standards AppliedMathematics Series 55, Washington, DC, 1964.2. R. C. Weast, Standard Mathematical Tables, the Chemical Rubber Co. (now CRCPress LLC), Cleveland, OH, 13th edition, 1964.)3. H. Flanders, R. R. Korfhage, and J.
J. Price, A First Course in Calculus with AnalyticGeometry, Academic Press, New York, 1973.© 2001 by CRC Press LLC.
