Ex12-22%20%20in%2016.1 - S E C T I O N 16.1 Integration in...

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S E C T I O N 16.1 Integration in Several Variables (ET Section 15.1) 843 4 0.4 0.4 0.6 0.8 0.6 0.6 3 0.8 1.6 1.8 2.5 2.1 0.8 2 0.5 1.5 3.2 3.5 2.1 0.6 1 0.4 0.8 1.3 1.5 1.4 0.6 0 0.3 0.3 0.5 0.8 0.5 0.4 y x 0 1 2 3 4 5 SOLUTION The function f ( x , y ) on the rectangle R = [0 , 5] × [0 , 4] gives the height of the mound at each point ( x , y ) R . We are given the values of f at some points. The volume is the double integral R f ( x , y ) d A , which we estimate by an average of four Riemann sums. The computations are fairly tedious and we will not show them here. The Riemann sum using the lower-left corner of each rectange is 27 . 4, for lower-right it is 27 . 8, for upper-right we get 28 . 3, and for upper-left we get 27 . 8 The average is 27 . 825. 12. Figure 18 shows a grid with values of f ( x , y ) at sample points in each square. Estimate the double integral of f ( x , y ) over the rectangles: (a) [ 0 . 25 , 1 ] × [ 0 . 5 , 1 ] (b) [− 0 . 5 , 0 . 5 ] × [ 0 , 1 ] (c) [− 1 , 0 . 5 ] × [− 1 , 0 . 5 ] y x 1 1 1 1 5.5 5.2 5.4 4.5 4.1 4.5 5 6 4 4.9 4 3.5 3.5 3.9 4.5 5 3.3 3.5 3.3 2.3 2.9 3.5 4.2 5 2 2.5 3 2 2.5 2.7 3.5 4.5 1.3 2 2.3 2.4 3 3.5 4.2 4.4 0 1 1 2.5 3 3.4 4 4.5 1 0 0.5 0 0 2 3 4 3 1.5 0.9 1 2 0 1 2 FIGURE 18 SOLUTION Each subrectangle is a square of side 0 . 25, hence its area is: A = 0 . 25 2 = 0 . 0625 We use the given data to estimate the double integral over the following squares.
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