CalcExercisesCh6

CalcExercisesCh6 - (Exercises for Section 6.1: Area) E.6.1...

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(Exercises for Section 6.1: Area) E.6.1 CHAPTER 6: APPLICATIONS OF INTEGRALS SECTION 6.1: AREA Assume that distances and lengths are measured in meters. 1) For parts a) and b) below, in the usual xy -plane … i) Sketch the region R bounded by the graphs of the given equations. Locate any intersection points of the graphs. ii) Set up the integral(s) for the area of R by integrating with respect to x . iii) Set up the integral(s) for the area of R by integrating with respect to y . iv) Find the area of the region by using either method from ii) or iii). ADDITIONAL PROBLEM : Find the area using the other method. a) y = ± x 2 and y = x 2 ± 8 b) y 2 = 4 ± x and x + 2 y = 1 2) In the tw -plane, sketch the regions bounded by the graphs of w = sin t and w = cos t , where t is restricted to the interval 0, 2 ± ² ³ ´ µ , and find the total area of the regions. 3) Find the area of the region bounded by the graphs of the given equations in the usual xy -plane. You do not have to sketch the region. a) y = 3 x 2 ± 5 and y = x 2 + 5 x ± 2 b) y = xx 2 + 16 , x = 0 , x = 3 , and y = 0 c) x ± y 3 = 0 and x + y + 2 y 2 = 0
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(Exercises for Section 6.2: Volumes of Solids of Revolution – Disks and Washers) E.6.2 SECTION 6.2: VOLUMES OF SOLIDS OF REVOLUTION – DISKS AND WASHERS Assume that distances and lengths are measured in meters. Assume that graphs are in the usual xy -plane, unless otherwise indicated. 1) The region R is bounded by the graphs of x + 2 y = 2 , x = 0 , and y = 0 . a) Sketch the region. b) Find the volume of the solid generated if R is revolved about the x -axis. c) Find the volume of the solid generated if R is revolved about the y -axis. 2) The region R is bounded by the graphs of y = ± x 2 and y = x 2 ± 8 . You should have sketched and found the area of this region in Section 6.1, Exercise 1a. a) Find the volume of the solid generated if R is revolved about the x -axis. b) Find the volume of the solid generated if R is revolved about the y -axis. 3) The region R in the tw -plane is bounded by the graphs of w = sin t , w = cos t , t = 0 , and t = ± 4 . (This region was related to Section 6.1, Exercise 2.) Find the volume of the solid generated if R is revolved about the t -axis. Hint: You will need to use a trig identity. 4) The region R in the xy -plane is bounded by the graphs of y = cos 2 x () , y = 0 , x = 0 , and x = 4 . Sketch R . Find the volume of the solid generated if R is revolved about the x -axis. Hint: You will need to use a trig identity.
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CalcExercisesCh6 - (Exercises for Section 6.1: Area) E.6.1...

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