SECTION 14.5 The Area Problem; The Integral 971 Now look at Table 6, which shows the approximations to the area under the graph of ( ) = f x x2 from 0 to 1 for = n 2, 4, 10, and 100 subintervals. Notice that the approximations to the actual area improve as the number of subintervals increases. Using left endpoints: n 2 4 10 100 Area 0.5 0.75 0.9 0.99 Using right endpoints: n 2 4 10 100 Area 1.5 1.25 1.1 1.01 Table 6 Figure 25 f(x) 5 x2 x y 5 15 25 4 subintervals; each of width 1 f(1) f(2) f(3) f(4) 5 0 1 2 3 4 Figure 26 f(x) 5 x2 x y 5 15 25 8 subintervals; each of width 1/2 f(1) f(1.5) f(2) f(4) f(4.5) 5 0 1 3 f(3) f(2.5) f(3.5) You are asked to confirm the entries in Table 6 in Problem 31. There is another useful observation about Example 1. Look again at Figures 24(a)–(d) and at Table 6. Since the graph of ( ) = f x x2 is increasing on [ ] 0, 1 , the choice of u as the left endpoint gives a lower-bound estimate to the actual area, while choosing u as the right endpoint gives an upper-bound estimate. Do you see why? Now Work PROBLEM 9 Approximating the Area under the Graph of ( ) = f x x2 Approximate the area under the graph of ( ) = f x x2 from 1 to 5: (a) Using four subintervals of equal width (b) Using eight subintervals of equal width In each case, choose the number u to be the left endpoint of each subinterval. Solution EXAMPLE 2 (a) See Figure 25. Using four subintervals of equal width, the interval [ ] 1, 5 is partitioned into subintervals of width − = 5 1 4 1 as follows: [ ] [ ] [ ] [ ] 1, 2 2, 3 3, 4 4, 5 Choosing u as the left endpoint of each subinterval, the area A under the graph of ( ) = f x x2 is approximated by ( ) ( ) ( ) ( ) ≈ ⋅ + ⋅ + ⋅ + ⋅ = + + + = A f f f f Area 1 1 2 1 3 1 4 1 1 4 9 16 30 (b) See Figure 26. Using eight subintervals of equal width, the interval [ ] 1, 5 is partitioned into subintervals of width − = 5 1 8 0.5 as follows: [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 1, 1.5 1.5, 2 2, 2.5 2.5, 3 3, 3.5 3.5, 4 4, 4.5 4.5, 5 Choosing u as the left endpoint of each subinterval, the area A under the graph of ( ) = f x x2 is approximated by [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ≈ ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + = + + + + + + + ⋅ = + + + + + + + ⋅ = A f f f f f f f f f f f f f f f f Area 1 0.5 1.5 0.5 2 0.5 2.5 0.5 3 0.5 3.5 0.5 4 0.5 4.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.5 1 2.25 4 6.25 9 12.25 16 20.25 0.5 35.5
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