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solutionExam 1_pdf - Version 040 EXAM 1 Odell(58340 This...

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Version 040 – EXAM 1 – Odell – (58340) 1 This print-out should have 19 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points When -2 -1 0 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 2 4 2 4 is the graph of a function f , use rectangles to estimate the definite integral I = integraldisplay 10 0 | f ( x ) | dx by subdividing [0 , 10] into 10 equal subin- tervals and taking right endpoints of these subintervals. 1. I 22 2. I 19 3. I 18 4. I 21 correct 5. I 20 Explanation: The definite integral I = integraldisplay 10 0 | f ( x ) | dx is the area between the graph of f and the interval [0 , 10]. The area is estimated using the gray-shaded rectangles in -2 -1 0 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 2 4 2 4 Each rectangle has base-length 1; and it’s height can be read off from the graph. Thus Area = 4 + 4 + 1 + 1 + 1 + 2 + 4 + 3 + 1 . Consequently, I 21 . 002 10.0 points Find an expression for the area of the region under the graph of f ( x ) = x 3 on the interval [1 , 8]. 1. area = lim n → ∞ n summationdisplay i =1 parenleftBig 1 + 10 i n parenrightBig 3 8 n 2. area = lim n → ∞ n summationdisplay i =1 parenleftBig 1 + 7 i n parenrightBig 3 8 n 3. area = lim n → ∞ n summationdisplay i =1 parenleftBig 1 + 7 i n parenrightBig 3 7 n correct 4. area = lim n → ∞ n summationdisplay i =1 parenleftBig 1 + 10 i n parenrightBig 3 7 n 5. area = lim n → ∞ n summationdisplay i =1 parenleftBig 1 + 8 i n parenrightBig 3 7 n 6. area = lim n → ∞ n summationdisplay i =1 parenleftBig 1 + 8 i n parenrightBig 3 8 n
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Version 040 – EXAM 1 – Odell – (58340) 2 Explanation: The area of the region under the graph of f on an interval [ a, b ] is given by the limit A = lim n → ∞ n summationdisplay i =1 f ( x i ) Δ x when [ a, b ] is partitioned into n equal subin- tervals [ a, x 1 ] , [ x 1 , x 2 ] , . . . , [ x n 1 , b ] each of length Δ x = ( b a ) /n and x i is an arbitrary sample point in [ x i 1 , x i ]. Consequently, when f ( x ) = x 3 , [ a, b ] = [1 , 8] , and x i = x i , we see that area = lim n → ∞ n summationdisplay i =1 parenleftBig 1 + 7 i n parenrightBig 3 7 n . 003 10.0 points Express the limit lim n → ∞ n summationdisplay i =1 5 x i sin x i Δ x as a definite integral on the interval [2 , 8]. 1. limit = integraldisplay 2 8 5 sin x dx 2. limit = integraldisplay 8 2 5 sin x dx 3. limit = integraldisplay 8 2 5 x sin x dx correct 4. limit = integraldisplay 2 8 5 x dx 5. limit = integraldisplay 2 8 5 x sin x dx 6. limit = integraldisplay 8 2 5 x dx Explanation: By definition, the definite integral I = integraldisplay b a f ( x ) dx of a continuous function f on an interval [ a, b ] is the limit I = lim n → ∞ n summationdisplay i =1 f ( x i ) Δ x of the Riemann sum n summationdisplay i =1 f ( x i ) Δ x formed when the interval [ a, b ] is divided into n subintervals of equal width Δ x and x i is any sample point in the i th subinterval [ x i 1 , x i ].
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