This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Version 099 EXAM 1 Radin (58415) 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 20 3. I 19 4. I 23 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 = 4 + 3 + 1 + 1 + 2 + 3 + 4 + 2 + 2 . Consequently, I 22 . 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 , 9]. 1. area = lim n n summationdisplay i = 1 parenleftBig 1 + 8 i n parenrightBig 3 8 n correct 2. area = lim n n summationdisplay i = 1 parenleftBig 1 + 11 i n parenrightBig 3 8 n 3. area = lim n n summationdisplay i = 1 parenleftBig 1 + 8 i n parenrightBig 3 9 n 4. area = lim n n summationdisplay i = 1 parenleftBig 1 + 9 i n parenrightBig 3 9 n 5. area = lim n n summationdisplay i = 1 parenleftBig 1 + 9 i n parenrightBig 3 8 n 6. area = lim n n summationdisplay i = 1 parenleftBig 1 + 11 i n parenrightBig 3 9 n Version 099 EXAM 1 Radin (58415) 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 3 , [ 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 3 8 n . 003 10.0 points Express the limit lim n n summationdisplay i =1 2 x i sin x i x as a definite integral on the interval [1 , 9]. 1. limit = integraldisplay 1 9 2 x dx 2. limit = integraldisplay 9 1 2 x dx 3. limit = integraldisplay 1 9 2 sin x dx 4. limit = integraldisplay 1 9 2 x sin x dx 5. limit = integraldisplay 9 1 2 x sin x dx correct 6. limit = integraldisplay 9 1 2 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...
View
Full
Document
This note was uploaded on 03/19/2008 for the course CH 301 taught by Professor Fakhreddine/lyon during the Spring '07 term at University of Texas at Austin.
 Spring '07
 Fakhreddine/Lyon

Click to edit the document details