SECTION 14.5 The Area Problem; The Integral 973 Figure 28 Solution Figure 28 shows the result using a TI-84 Plus CE graphing calculator. Consult the user’s manual for the proper keystrokes. The area under the graph of ( ) = f x x2 from 1 to 5 is 124 3 . In calculus, techniques are given for evaluating integrals to obtain exact answers algebraically. ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 14.5 Assess Your Understanding 1. The formula for the area A of a rectangle of length l and width w is . (p. A15) 2. ∑( ) + = = k2 1 k 1 4 . (pp. 859–862) In Problems 5 and 6, refer to the figure. The interval [ ] 1, 3 is partitioned into two subintervals [ ] 1, 2 and [ ] 2, 3 . y 5 f(x) 1 2 f(1) 5 1, f(2) 5 2, f(3) 5 4 f(3) 5 4 f(2) 5 2 f(1) 5 1 3 x y A 5. Approximate the area A , choosing u as the left endpoint of each subinterval. 6. Approximate the area A , choosing u as the right endpoint of each subinterval. In Problems 7 and 8, refer to the figure. The interval [ ] 0, 8 is partitioned into four subintervals ] [ ] [ [ ] 0,2, 2,4, 4,6, and [ ] 6, 8 . y 5 f(x) 2 4 6 8 f(0) 5 10, f(2) 5 6, f(4) 5 7, f(6) 5 5, f(8) 5 1 10 5 x y 7. Approximate the area A , choosing u as the left endpoint of each subinterval. 8. Approximate the area A , choosing u as the right endpoint of each subinterval. 9. The function ( ) = f x x3 is defined on the interval [ ] 0, 6 . (a) Graph f. In (b)–(e), approximate the area A under f from 0 to 6 as follows: (b) Partition [ ] 0, 6 into three subintervals of equal width and choose u as the left endpoint of each subinterval. (c) Partition [ ] 0, 6 into three subintervals of equal width and choose u as the right endpoint of each subinterval. (d) Partition [ ] 0, 6 into six subintervals of equal width and choose u as the left endpoint of each subinterval. (e) Partition [ ] 0, 6 into six subintervals of equal width and choose u as the right endpoint of each subinterval. (f) What is the actual area A ? 10. Repeat Problem 9 for ( ) = f x x4 . 11. The function ( ) = − + f x x3 9 is defined on the interval [ ] 0, 3 . (a) Graph f. In (b)–(e), approximate the area A under f from 0 to 3 as follows: (b) Partition [ ] 0, 3 into three subintervals of equal width and choose u as the left endpoint of each subinterval. (c) Partition [ ] 0, 3 into three subintervals of equal width and choose u as the right endpoint of each subinterval. (d) Partition [ ] 0, 3 into six subintervals of equal width and choose u as the left endpoint of each subinterval. (e) Partition [ ] 0, 3 into six subintervals of equal width and choose u as the right endpoint of each subinterval. (f) What is the actual area A ? Skill Building 3. The integral from a to b of ( ) f x is denoted by . 4. Multiple Choice For a continuous, nonnegative function f , the area under the graph of f from a to b is denoted by . (a) ( ) ( ) − − f b f a b a (b) ∫ ( ) f x dx a b (c) ( ) ( ) ′ − ′ f b f a (d) − ⋅ Δ b a n x Concepts and Vocabulary 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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