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Unformatted text preview: Version 071 EXAM 1 Gompf (58370) 1 This printout should have 19 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 10.0 points When 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  f ( x )  dx by subdividing [0 , 10] into 10 equal subin tervals and taking right endpoints of these subintervals. 1. I 21 2. I 19 3. I 18 4. I 20 5. I 22 correct Explanation: The definite integral I = integraldisplay 10  f ( x )  dx is the area between the graph of f and the interval [0 , 10]. The area is estimated using the grayshaded rectangles in 2 4 6 8 10 2 4 2 4 Each rectangle has baselength 1; and its height can be read off from the graph. Thus Area = 5 + 4 + 2 + 1 + 1 + 2 + 3 + 3 + 1 . Consequently, I 22 . 002 10.0 points Find an expression for the area of the region under the graph of f ( x ) = x 5 on the interval [1 , 9]. 1. area = lim n n summationdisplay i = 1 parenleftBig 1 + 11 i n parenrightBig 5 9 n 2. area = lim n n summationdisplay i = 1 parenleftBig 1 + 8 i n parenrightBig 5 9 n 3. area = lim n n summationdisplay i = 1 parenleftBig 1 + 9 i n parenrightBig 5 9 n 4. area = lim n n summationdisplay i = 1 parenleftBig 1 + 8 i n parenrightBig 5 8 n correct 5. area = lim n n summationdisplay i = 1 parenleftBig 1 + 11 i n parenrightBig 5 8 n 6. area = lim n n summationdisplay i = 1 parenleftBig 1 + 9 i n parenrightBig 5 8 n Version 071 EXAM 1 Gompf (58370) 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 ) 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 x = ( b a ) /n and x i is an arbitrary sample point in [ x i 1 , x i ]. Consequently, when f ( x ) = x 5 , [ a, b ] = [1 , 9] , and x i = x i , we see that area = lim n n summationdisplay i = 1 parenleftBig 1 + 8 i n parenrightBig 5 8 n . 003 10.0 points Express the limit lim n n summationdisplay i =1 5 x i sin x i x as a definite integral on the interval [1 , 8]. 1. limit = integraldisplay 1 8 5 sin x dx 2. limit = integraldisplay 8 1 5 x dx 3. limit = integraldisplay 1 8 5 x sin x dx 4. limit = integraldisplay 1 8 5 x dx 5. limit = integraldisplay 8 1 5 x sin x dx correct 6. limit = integraldisplay 8 1 5 sin 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 ) x of the Riemann sum n summationdisplay i = 1 f ( x i ) x formed when the interval [ a, b ] is divided into n subintervals of equal width x and x i is any sample point in the i th subinterval [ x i...
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This note was uploaded on 12/04/2009 for the course M 408L taught by Professor Radin during the Spring '08 term at University of Texas at Austin.
 Spring '08
 RAdin
 Calculus

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