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Section6_6 - Section 6.6 Estimating denite integrals In...

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(3/20/08) Section 6.6 Estimating definite integrals In this section we discuss techniques for finding approximate values of definite integrals and work with applications where the data is given approximately by graphs and tables. We also present the Trapezoid and Simpson’s Rules for approximating integrals, discuss upper and lower Riemann sums, and give error estimates for the Midpoint, Trapezoid, and Simpson’s Rules. Topics: Finding approximate integrals from graphs and tables The Trapezoid Rule Upper and lower Riemann sums Simpson’s Rule Error estimates Finding approximate integrals from graphs and tables The St. Francis dam, constructed in 1928 northeast of the present Magic Mountain near Los Angeles, was designed by William Mulholland with the plans he had used for the Mulholland dam that still supports the Hollywood Reservoir. The sides of the canyon where the St. Francis dam was built had geological flaws that were not recognized at the time. When the reservoir was filled for the first time, the dam broke, flooding the San Francisquito and Santa Clara River valleys and drowning 450 people. This tragedy ended the previously glamorous career of the self-educated engineer Mulholland, who had been the chief architect of the Los Angeles–Owens River Aqueduct that supplies much of the water to Los Angeles. Example 1 Figure 1 shows the graph of the rate of flow of water r = r ( t ) from the St. Francis dam in a 90-minute period starting 15 minutes before it broke. (1) Estimate the total volume of water that flowed from the dam for 0 t 90. t 15 30 45 60 75 90 r (thousand acre-feet per minute) 20 40 60 r = r ( t ) (minutes) t 15 30 45 60 75 90 r (thousand acre-feet per minute) 20 40 60 r = r ( t ) (minutes) FIGURE 1 FIGURE 2 Solution Suppose that V ( t ) is the volume of water that has flowed from the dam from time 0 to time t 0. Then V prime ( t ) = r ( t ), and since V (0) = 0, Part I of the Fundamental Theorem in Section 6.3 shows that the volume of water to flow from the dam for 0 t 90 is V (90) = V (90) - V (0) = integraldisplay 90 0 r ( t ) dt. Since r = r ( t ) is a positive function, this integral is equal to the area of the region between its graph and the t -axis for 0 t 90 in Figure 1. (1) Data adapted from “A man, a dam and a disaster: Mulholland and the St.Francis Dam” by J. Rogers, Ventura County Historical Society Quarterly , Vol. 77, Ventura California: Ventura County Historical Society, 1995, p. 76. 225
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p. 226 (3/20/08) Section 6.6, Estimating definite integrals We estimate this area by six rectangles whose sides are determined by the vertical lines at t = 0 , 15 , 30 , 45 , 60 , 75, and 90. We approximate each portion of the curved region by a rectangle, as in Figure 2, with the tops chosen to have the area of each rectangle appear approximately equal to the area of the corresponding portion of the region under the curve. The width of each rectangle is 15. From the sketch we estimate the heights of the rectangles to be 2, 50, 40, 23, 11, and 4, so that [Total volume] (2)(15) + (50)(15) + (40)(15) + (23)(15) + (11)(15) + (4)(15) = 1950 thousand acre-feet .
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