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### solution_c6pdf

Course: M 408S, Spring 2011
School: University of Texas
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Word Count: 804

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(kma786) armington 6.1 Stepp (55860) This print-out should have 6 questions. Multiple-choice questions may continue on the next column or page nd all choices before answering. 001 10.0 points y x 1 Express the area, A, between the graph of f and the x-axis on the interval [0, 4] in terms of denite integral(s) when f ( x) = 2 x x2 . In terms of denite integrals, therefore, the required area is given by 2 4...

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(kma786) armington 6.1 Stepp (55860) This print-out should have 6 questions. Multiple-choice questions may continue on the next column or page nd all choices before answering. 001 10.0 points y x 1 Express the area, A, between the graph of f and the x-axis on the interval [0, 4] in terms of denite integral(s) when f ( x) = 2 x x2 . In terms of denite integrals, therefore, the required area is given by 2 4 A= 0 3 4 (2x x2 ) dx 002 2 (2x x2 ) dx . 1. A = 3 0 (2x x2 ) dx + 3 4 (2x x2 ) dx (2x x2 ) dx 4 10.0 points Find the area enclosed by the graphs of f (x) = cos x , on [0, ]. 1. area = 4( 2 + 1) 2. area = 2 2 correct 3. area = 2 4. area = 2( 2 + 1) 5. area = 2 + 1 6. area = 4 2 Explanation: The area between the graphs of y = f (x) and y = g (x) on the interval [a, b] is expressed by the integral b 2. A = 0 (2x x2 ) dx 2 3 g (x) = sin x 3. A = 0 correct 4 (2x x2 ) dx 2 (2x x2 ) dx 4. A = 0 (2x x2 ) dx 1 4 5. A = 4 0 (2x x2 ) dx + 1 (2x x2 ) dx 6. A = 0 1 (x2 2x) dx 4 7. A = 0 (2x x2 ) dx 2 1 (2x x2 ) dx 4 8. A = 0 (2x x ) dx + 2 2 (2x x2 ) dx A= a |f (x) g (x)| dx , which for the given functions is the integral Explanation: The graph of f is a parabola opening downwards and crossing the x-axis at x = 0 and x = 2. Thus the required area is similar to the shaded region in the A= 0 |cos x sin x| dx . But, as the graphs armington (kma786) 6.1 Stepp (55860) y 2. Area = 3. Area = /2 cos : sin : 4. Area = 5. Area = 35 sq. units 3 41 sq. units 3 38 sq. units 3 29 sq. units 3 2 of y = cos x and y = sin x on [0, ] show, cos sin Thus /4 0, 0, on [0, /4], on [/4, ]. Explanation: The graph of f is a parabola opening upwards, while the graph of g is a straight line of slope 3. Both graphs pass through the origin. Thus the required area is the shaded region in the gure below graph of f A= 0 {cos sin } d {cos sin } d = A1 A2 . graph of g /4 But by the Fundamental Theorem of Calculus, A1 = while A2 = sin + cos /4 4 sin + cos /4 0 = 21, To express the area as a denite integral we need to nd where the graphs intersect, i.e. when x2 x = 3 x . As the solutions of this equation are x = 0 and x = 4, the area of the shaded region is thus given by 4 = (1 + 2) . Consequently, area = A1 A2 = 2 2 . 003 10.0 points Area = 0 4 (g (x) f (x)) dx (4x x2 ) dx = 1 2 x2 3 x3 . 0 4 Find the bounded area enclosed by the graphs f of (x) = x2 x, 1. Area = g ( x) = 3 x . = 0 Consequently, Area = 32 sq.units . 3 32 sq. units correct 3 armington (kma786) 6.1 Stepp (55860) (not drawn to scale). Thus 004 10.0 points 1 3 Find the area of the bounded region enclosed by the graphs of f (y ) = y 2 , g ( y ) = 3 2y . I= 3 3 2y y 2 dy = 1 3y y y 3 3 2 1 . 3 1. Area = 11 sq. units 32 2. Area = sq. units correct 3 3. Area = 34 sq. units 3 Consequently Area = 32 sq. units . 3 10.0 points 005 Find the area enclosed by the graphs of x + y = 5 , x y = 0 , y + 4x = 5 . 1. Area = 4 sq. units 2. Area = 15 sq. units correct 4 4. Area = 10 sq. units 5. Area = 31 sq. units 3 Explanation: The area of the enclosed region is given by the integral b 3. Area = 3 sq. units {g (y ) f (y )} dy 4. Area = 5. Area = 7 sq. units 2 13 sq. units 4 I= a where a, b are the y -coordinates of the points of intersection of the two graphs. Now a, b are the solutions of the equation f (y ) = g (y ), i.e., y + 2y 3 = (y + 3)(y 1) = 0 . Graphically, therefore, we have to nd the area of the shaded region in y 2 Explanation: Each of the graphs is a straight line, so the enclosed region is the shaded triangle in y 1 x 3 x armington (kma786) 6.1 Stepp (55860) To express the area as an integral we need to nd where the lines intersect. Now the line x y = 0 intersects x+ y = 5, at the points ( 1, 1 ) , 55 , 22 y + 4x = 5 4 Consider the following possible values for the area of these regions: 4 I. II. III. A1 = 1 {g (x) f (x)} dx, 4 A2 = 9 4 g (x) dx, 1 A3 = 1 respectively. Thus the area of the shaded triangle is given by 1 0 f (x) dx 3. Which are correct without further restrictions on f and g ? 1. I only 2. II only 3. I and II only ((5 x) (5 4x)) dx 5 2 + 1 1 ((5 x) (x)) dx 5 2 = 0 3x dx + 1 (5 2x) dx . 4. none are correct correct 5. I and III only 6. all of them 7. III only Consequently, the shaded triangle has Area = 15 sq.units . 4 keywords: area between curves, linear functions 006 10.0 points 8. II and III only Explanation: Given functions y = F (x) and y = G(x) with a x b, the area between their graphs as functions of x can be written as b The graphs of functions y = f ( x) , y = g ( x) determine regions A1 , A2 , A3 in a rectangle in the rst quadrant as shown in area = a |F (x) G(x)| dx . Thus for the particular functions f, g dened by their graphs, we see that (4, 3) A3 A1 A2 (1, 1) f: g: I. II. III. False. False. False.
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