970 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function In approximating the area under the graph of a function f from a to b, the choice of the number u in each subinterval is arbitrary. For convenience, we shall always pick u as either the left endpoint of each subinterval or the right endpoint. The choice of how many subintervals to use is also arbitrary. In general, the more subintervals used, the better the approximation will be. Let’s look at a specific example. Figure 24 f(0) f(x) 5 2x x y 2 1 (a) 2 subintervals; u’s are left endpoints 2 1 1 0 f ( ) 2 1 f(0) f(x) 5 2x x y 2 1 (c) 4 subintervals; u’s are left endpoints 1 0 4 3 2 1 4 1 f ( ) 4 1 f ( ) 4 3 f ( ) 2 1 f(1) f(x) 5 2x x y 2 1 (b) 2 subintervals; u’s are right endpoints 1 0 2 1 f ( ) 2 1 f(1) f(x) 5 2x x y 2 1 (d) 4 subintervals; u’s are right endpoints 1 0 2 1 4 3 4 1 f ( ) 4 3 f ( ) 2 1 f ( ) 4 1 ( ) ( ) = ⋅ = = ⋅ = f f 0 2 0 0; 1 2 2 1 2 1 Approximating the Area under the Graph of ( ) = f x x2 from 0 to 1 Approximate the area A under the graph of ( ) = f x x2 from 0 to 1 as follows: (a) Partition [ ] 0, 1 into two subintervals of equal width and choose u as the left endpoint. (b) Partition [ ] 0, 1 into two subintervals of equal width and choose u as the right endpoint. (c) Partition [ ] 0, 1 into four subintervals of equal width and choose u as the left endpoint. (d) Partition [ ] 0, 1 into four subintervals of equal width and choose u as the right endpoint. (e) Compare the approximations found in parts (a)–(d) with the actual area. Solution (a) Partition [ ] 0, 1 into two subintervals, each of width 1 2 , and choose u as the left endpoint. See Figure 24(a). The area A is approximated as ( ) ( ) ≈ ⋅ + ⋅ = ⋅ + ⋅ = = A f f 0 1 2 1 2 1 2 0 1 2 1 1 2 1 2 0.5 (b) Partition [ ] 0, 1 into two subintervals, each of width 1 2 , and choose u as the right endpoint. See Figure 24(b). The area A is approximated as ( ) ( ) ≈ ⋅ + ⋅ = ⋅ + ⋅ = = A f f 1 2 1 2 1 1 2 1 1 2 2 1 2 3 2 1.5 (c) Partition [ ] 0, 1 into four subintervals, each of width 1 4 , and choose u as the left endpoint. See Figure 24(c). The area A is approximated as ( ) ( ) ( ) ( ) ≈ ⋅ + ⋅ + ⋅ + ⋅ =⋅+⋅+⋅+⋅= = A f f f f 0 1 4 1 4 1 4 1 2 1 4 3 4 1 4 0 1 4 1 2 1 4 1 1 4 3 2 1 4 3 4 0.75 (d) Partition [ ] 0, 1 into four subintervals, each of width 1 4 , and choose u as the right endpoint. See Figure 24(d). The area A is approximated as ( ) ( ) ( ) ( ) ≈ ⋅ + ⋅ + ⋅ + ⋅ =⋅+⋅+⋅+⋅= = A f f f f 1 4 1 4 1 2 1 4 3 4 1 4 1 1 4 1 2 1 4 1 1 4 3 2 1 4 2 1 4 5 4 1.25 (e) The actual area under the graph of ( ) = f x x2 from 0 to 1 is the area of a right triangle whose base is of length 1 and whose height is 2.The actual area A is therefore = × = ⋅ ⋅ = A 1 2 base height 1 2 1 2 1 EXAMPLE 1
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