972 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function Steps for Approximating the Area A Under the Graph of a Function ( ) = y f x from a to b Step 1 Partition the interval [ ] a b , into n subintervals of equal width. The width Δx of each subinterval is then Δ = − x b a n Step 2 In each subinterval, pick a number u and evaluate the function f at each u. This results in n numbers … u u u , , , n 1 2 and n functional values ( ) ( ) ( ) f u f u f u , , , . n 1 2 … Step 3 Form n rectangles with width equal to Δx, the width of each subinterval, and with length equal to the functional value ( ) = f u i n , 1, 2, , . i … See Figure 27. Step 4 Add the areas of the n rectangles. ∑ ( ) ( ) ( ) ( ) +++= Δ+ Δ++ Δ = Δ = A A A f u x f u x f u x f u x n n i n i 1 2 1 2 1 This number is the approximation to the area A under the graph of f from a to b. Figure 27 y 5 f(x) x y a b Dx Dx Dx Dx u1 u3 un u2 f(u1) f(u2) f(u3) f(un) Definition of Area We have observed that the larger the number n of subintervals used, the better the approximation to the area under the graph of f from a to b . If we let n become unbounded, we obtain the exact area under the graph of f from a to b . DEFINITION Area under the Graph of a Function from a to b Suppose a function f is nonnegative and continuous on a closed interval [ ] a b , . Partition [ ] a b , into n subintervals, each of width Δ = − x b a n . In each subinterval, choose a number = … u i n , 1, 2, , , i and evaluate ( ) f u . i Form the products ( )Δ f u x i and add them, obtaining the sum ∑ ( )Δ = f u x i n i 1 If the limit of this sum exists as →∞ n , that is, if ∑ ( )Δ →∞ = f u x lim n i n i 1 exists, it is defined as the area under the graph of f from a to b and is denoted by ∫ ( ) f x dx a b which is read as “the integral from a to b of ( ) f x . ” 2 Approximate Integrals Using a Graphing Utility Using a Graphing Utility to Approximate an Integral Use a graphing utility to approximate the area under the graph of ( ) = f x x2 from 1 to 5. That is, evaluate the integral ∫ x dx 1 5 2 EXAMPLE 3
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