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Unformatted text preview: Massaro, Michael Homework 4 Due: Sep 25 2007, 3:00 am Inst: Shinko Harper 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 ( x + 1) 1 2 x dx ? 1. 2 4 6 2 2 4 6 2 2. 2 4 6 2 2 4 6 2 correct 3. 2 4 6 2 2 2 4 6 4. 2 4 6 2 2 4 6 5. 2 4 6 2 2 2 4 6 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 ) = x + 1 , g ( x ) = 1 2 x, therefore, the value of Z 4 n ( x + 1) 1 2 x o dx is the area of the shaded region 2 4 6 2 2 4 6 2 . keywords: integral, region, area Massaro, Michael Homework 4 Due: Sep 25 2007, 3:00 am Inst: Shinko Harper 2 002 (part 1 of 1) 10 points Find the area of the region enclosed by the graphs of f ( x ) = 16 x 2 , g ( x ) = x + 3 , on the interval [0 , 1]. 1. Area = 23 2 sq. units 2. Area = 12 sq. units 3. Area = 71 6 sq. units 4. Area = 73 6 sq. units correct 5. Area = 35 3 sq. units Explanation: The graph of f is a parabola opening down wards and crossing the xaxis at x = 4, while the graph of g is a straight line with slope 1 and yintercept at y = 3. Now on [0 , 1] we see that f ( x ) = 16 x 2 x + 3 = 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 16 x 2 x 3 o dx = 16 x 1 3 x 3 1 2 x 2 3 x 1 . Consequently, Area = 73 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 4 x, g ( x ) = 8 x 2 x 2 . 1. area = 65 2 sq.units 2. area = 32 sq.units correct 3. area = 61 2 sq.units 4. area = 63 2 sq.units 5. area = 31 sq.units Explanation: The graph of f is a parabola opening up wards and crossing the xaxis at x = 0 and x = 4, while the graph of g is a parabola opening downwards and crossing the xaxis at x = 0 and x = 4. 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 4 ( g ( x ) f ( x )) dx = Z 4 (12 x 3 x 2 ) dx. Massaro, Michael Homework 4 Due: Sep 25 2007, 3:00 am Inst: Shinko Harper 3 Now Z 4 (12 x 3 x 2 ) dx = h 6 x 2 x 3 i 4 = 32 . Thus Area = 32 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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 Fall '08
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