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Unformatted text preview: Valenica, Daniel Homework 4 Due: Sep 25 2007, 3:00 am Inst: Cheng 1 This printout should have 22 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points For which one of the following shaded re gions is its area represented by the integral Z 4 (6 x ) 1 2 x dx ? 1. 2 4 6 2 2 4 6 2 2. 2 4 6 2 2 2 4 6 3. 2 4 6 2 2 2 4 6 4. 2 4 6 2 2 4 6 2 5. 2 4 6 2 2 4 6 correct Explanation: If f ( x ) g ( x ) for all x in an interval [ a, b ], then the area between the graphs of f and g is given by Area = Z b a n f ( x ) g ( x ) o dx. When f ( x ) = 6 x, g ( x ) = 1 2 x, therefore, the value of Z 4 n (6 x ) 1 2 x o dx is the area of the shaded region 2 4 6 2 2 4 6 . keywords: integral, region, area Valenica, Daniel Homework 4 Due: Sep 25 2007, 3:00 am Inst: Cheng 2 002 (part 1 of 1) 10 points Find the area of the region enclosed by the graphs of f ( x ) = 9 x 2 , g ( x ) = x + 1 , on the interval [0 , 1]. 1. Area = 22 3 sq. units 2. Area = 7 sq. units 3. Area = 41 6 sq. units 4. Area = 20 3 sq. units 5. Area = 43 6 sq. units correct Explanation: The graph of f is a parabola opening down wards and crossing the xaxis at x = 3, while the graph of g is a straight line with slope 1 and yintercept at y = 1. Now on [0 , 1] we see that f ( x ) = 9 x 2 x + 1 = g ( x ) , so the area between the graphs of f and g on [0 , 1] is given by Area = Z 1 n f ( x ) g ( x ) o dx = Z 1 n 9 x 2 x 1 o dx = 9 x 1 3 x 3 1 2 x 2 1 x 1 . Consequently, Area = 43 6 sq. units . keywords: integral, area 003 (part 1 of 1) 10 points Find the area bounded by the graphs of f and g when f ( x ) = x 2 3 x, g ( x ) = 6 x 2 x 2 . 1. area = 15 sq.units 2. area = 29 2 sq.units 3. area = 14 sq.units 4. area = 31 2 sq.units 5. area = 27 2 sq.units correct Explanation: The graph of f is a parabola opening up wards and crossing the xaxis at x = 0 and x = 3, while the graph of g is a parabola opening downwards and crossing the xaxis at x = 0 and x = 3. Thus the required area is the shaded region in the figure below graph of g graph of f (graphs not drawn to scale). In terms of definite integrals, therefore, the required area is given by Area = Z 3 ( g ( x ) f ( x )) dx = Z 3 (9 x 3 x 2 ) dx. Valenica, Daniel Homework 4 Due: Sep 25 2007, 3:00 am Inst: Cheng 3 Now Z 3 (9 x 3 x 2 ) dx = h 9 2 x 2 x 3 i 3 = 27 2 . Thus Area = 27 2 sq.units . keywords: definite integral, area between graphs, quadratic functions 004 (part 1 of 3) 10 points The shaded region in is bounded by the graphs of f ( x ) = 1 + x x 2 x 3 and g ( x ) = 1 x....
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This homework help was uploaded on 04/16/2008 for the course CALC 303L taught by Professor Cheng during the Fall '07 term at University of Texas at Austin.
 Fall '07
 CHENG

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