SalasSV_09_09_ex - 9.9 THE AREA OF A SURFACE OF REVOLUTION...

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EXERCISES 9.9 ±ind the length of the curve, locate the centroid, and determine the area of the surface generated by revolving the curve about the x -axis. 1. f ( x ) = 4, x [0, 1]. 2. f ( x ) = 2 x , x [0, 1]. 3. y = 4 3 x , x [0, 3]. 4. y =− 12 5 x + 12, x [0, 5]. 5. x ( t ) = 3 t , y ( t ) = 4 t ; t [0, 2]. 6. r = 5, θ [0, 1 4 π ]. 7. x ( t ) = 2 cos t , y ( t ) = 2 sin t ; t [0, 1 6 π ]. 8. x ( t ) = cos 3 t , y ( t ) = sin 3 t ; t [0, 1 2 π ]. 9. x 2 + y 2 = a 2 , x [ 1 2 a , 1 2 a ], y > 0, a > 0. 10. r = 1 + cos θ , θ [0, π ]. ±ind the area of the surface generated by revolving the curve about the x -axis. 11. f ( x ) = 1 3 x 3 , x [0, 2]. 12. f ( x ) = x , x [1, 2]. 13. 4 y = x 3 , x [0, 1]. 14. y 2 = 9 x , x [0, 4]. 15. y = cos x , x [0, 1 2 π ]. 16. f ( x ) = 2 1 x , x [ 1, 0]. 17. r = e θ , θ [0, 1 2 π ]. 18. y = cosh x , x [0, ln 2]. 19. Take a > 0. One arch of a cycloid can be deFned parametri- cally by setting x ( θ ) = a ( θ sin θ ), y ( θ ) = a (1 cos θ ), Letting θ range from 0 to 2 π . (a) ±ind the area under the curve. (b) ±ind the area of the surface generated by revolving the arch about the x -axis. ± c 20. Take a > 0. The curve x ( θ ) = 3 a cos θ + a cos 3 θ , y ( θ ) = 3 a sin θ a sin 3 θ , with θ ranging from 0 to 2 π , is called a hypocycloid . (a) Use a graphing utility to draw the curve. (b) ±ind the area enclosed by the curve. (c) Set up a deFnite integral that gives the area of the surface generated by revolving the curve about the x -axis. 21. By cutting a cone of slant height s and base radius r along a lateral edge and laying the surface ²at, we can form a sector of a circle of radius s . (See the Fgure.) Use this idea to verify ±ormula (9.9.1). r 2 r π s s , in radians area = 1 2 2 θ s 22. The Fgure shows a ring formed by two quarter-circles. Call the corresponding quarter-discs ± a and ± r . By Section 6.4, ± a has centroid (4 a / 3 π ,4 a / 3 π ) and ± r has centroid (4 r / 3 π r / 3 π ). y x a r r a
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582 ± CHAPTER 9 THE CONIC SECTIONS; POLAR COORDINATES; PARAMETRIC EQUATIONS (a) Without integration, calculate the centroid of the ring. (b) Find the centroid of the outer arc from your answer to part (a) by letting a tend to r . 23. (a) Find the centroid of each side of the triangle in the ±gure. y x 5 4 3 (b) Use your answers in part (a) to calculate the centroid of the triangle.
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SalasSV_09_09_ex - 9.9 THE AREA OF A SURFACE OF REVOLUTION...

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