Math 132 Fall 2007 Exam III 1 The region in the first quadrant that is bounded above by y 4 x x 2 and bounded below by y x is rotated about the vertical line x 1 What is the volume of the resulting solid of revolution 15 2 j 15 a b 16 c 18 d 21 e 45 2 f 20 g 27 2 Solution e First we plot the planar region that is to be rotated f x 4 x x 2 g x x f x 4 x x2 g x x The planar region bounded by y x and y 4 x x2 is shown below Problem 1 Figure 1 h 25 2 i 14 When this region is rotated about the specified axis the following solid results Problem 1 Figure 2 Calculation of the Volume by the Method of Cylindrical Shells The next figure shows a typical cylindrical shell Problem 1 Figure 3 Here is the calculation of the volume by means of the Method of Cylindrical Shells radius of shell x x 1 height of shell x f x g x radius of shell x x 1 height of shell x f x g x Volume Int 2 Pi radius of shell x height of shell x x 0 3 3 Volume 2 x 1 3 x x2 dx 0 Volume 2 Pi Int expand radius of shell x height of shell x x 0 3 3 Volume 2 2 x2 x3 3 x dx 0 antiderivative int 2 x 2 x 3 3 x x antiderivative 2 x3 3 x4 4 3 x2 2 Volume simplify subs x 3 antiderivative subs x 0 antiderivative Volume 45 4 Calculation of the Volume by the Method of Washers It is possible but more difficult to obtain the volume using washers A glance at Figure 2 above indicates that we have to divide the integration into two pieces one for 0 y 3 and one for 3 y 4 Figures 4 and 5 below show the two types of washer For y 3 the outer radius of the washer extends to the curve y x For 3 y the outer radius of the washer extends to the curve y 4 x x2 Problem 1 Figure 4 Problem 1 Figure 5 Volume Int Pi 1 y 2 1 2 sqrt 4 y 2 y 0 3 Int Pi 1 2 sqrt 4 y 2 1 2 sqrt 4 y 2 y 3 4 3 2 Volume 1 y 3 0 4 2 4 y dy 3 4 y 3 2 2 4 y dy 3 Volume int Pi 1 y 2 1 2 sqrt 4 y 2 y 0 3 int Pi 1 2 sqrt 4 y 2 1 2 sqrt 4 y 2 y 3 4 Volume 45 2 2 The region in the first quadrant bounded above by y x and below by y x2 for 0 x 1 is rotated about the line y 2 What is the volume of the solid of revolution that results 8 15 f 2 a 3 5 3 g 8 b 2 3 5 h 8 c 4 5 3 i 4 d 14 15 7 j 8 e Solution a The region shown with blue boundary is rotated about y 2 In the next figure we show the solid that results from the rotation The inner boundaty in the shape of a conical frustum is rendered with solid tan The outer boundary a paraboloid is rendered using a brown wireframe A washer is also shown Calculation of the Volume by the Method of Washers The outer radius is the distance from y x2 to y 2 namely 2 x2 The inner radius is the distance from y x to y 2 namely 2 x volume Pi int 2 x 2 2 2 x 2 x 0 1 volume 8 15 Calculation of the Volume by the Method of Cylindrical Shells For a shell that has an edge at level y the radius is 2 y and the height of the shell is int 2 Pi 2 y sqrt y y y 0 1 8 15 3 x 1 1 y for x 1 3 4x 2 3 Calculate the arc length of the graph of a 5 12 b 11 24 c 1 2 d 13 24 f 5 8 g 2 3 h 17 24 i 3 4 e j 7 12 5 6 Solution d f x x 3 3 1 4 x f x 1 3 x3 11 4 x diff f x x 2 2 1 x 4 x2 2 expand 1 diff f x x 2 1 x4 1 2 16 x4 eqn 1 diff f x x 2 x 2 1 4 x 2 2 2 2 1 2 1 x eqn 1 x2 2 2 4x 4x testeq eqn true arc length Int x 2 1 4 x 2 x 1 2 1 int x 2 1 4 x 2 x 1 2 1 y y 1 2 1 13 arc length x d x 24 4 x2 1 2 4 The line segment with end points 2 3 and 6 0 is rotated about the y axis What is the surface area of the resulting conical frustum a 12 f 27 b 16 g 28 c 20 h 32 d 24 i 36 e 25 j 40 Solution j 2 2 6 or 8 The slant length is 32 42 2 or 5 The surface area is the product of the average circumference and the slant length 40 The average circumference of the frustum is 5 The graph of y 3 x for 4 x 10 is rotated about the x axis What is the surface area of the resulting figure a 54 f 84 b 60 g 94 c 64 h 102 Solution i f x 3 sqrt x d 72 i 109 e 80 j 115 f x 3 x Surface area 2 Pi Int f x sqrt 1 diff f x x 2 x 4 10 10 3 Surface area 2 4 x 9 x 2 dx 4 Answer Pi Int 3 4 x 9 1 2 x 4 10 10 Answer 3 4 x 9 dx 4 Answer student changevar u 4 x 9 Answer u 49 3 u Answer du 4 25 Surface area value Answer Surface area 109 6 Let R be the trapezoidal region in the first quadrant that is bounded above by the graph of y 2 x below by the x axis on the left by the vertical line x 1 and on the right by the vertical line x 2 What is the moment of R about the vertical line x 1 Assume that R has a constant density equal to 1 a 20 3 b 7 f 9 g 32 3 23 3 d 8 h 11 i 35 3 c e 25 3 j 12 Solution c f x 2 x f x 2 x Moment x 1 Int x 1 f x x 1 2 2 Momentx 1 2 x 1 x dx 1 Moment …
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