SalasSV_09_08_ex

# SalasSV_09_08_ex - 9.8 ARC LENGTH AND SPEED 573 EXERCISES...

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9.8 ARC LENGTH AND SPEED ± 573 EXERCISES 9.8 Find the length of the graph and compare it to the straight-line distance between the endpoints of the graph. 1. f ( x ) = 2 x + 3, x [0, 1]. 2. f ( x ) = 3 x + 2, x [0, 1]. 3. f ( x ) = ( x 4 9 ) 3 / 2 , x [1, 4]. 4. f ( x ) = x 3 / 2 , x [0, 44]. 5. f ( x ) = 1 3 x ( x 3), x [0, 3]. 6. f ( x ) = 2 3 ( x 1) 3 / 2 , x [1, 2]. 7. f ( x ) = 1 3 ( x 2 + 2) 3 / 2 , x [0, 1]. 8. f ( x ) = 1 3 ( x 2 2) 3 / 2 , x [2, 4]. 9. f ( x ) = 1 4 x 2 1 2 ln x , x [1, 5]. 10. f ( x ) = 1 8 x 2 ln x , x [1, 4]. 11. f ( x ) = 3 8 x 4 / 3 3 4 x 2 / 3 , x [1, 8]. 12. f ( x ) = 1 10 x 5 + 1 6 x 3 , x [1, 2]. 13. f ( x ) = ln ( sec x ), x [0, 1 4 π ]. 14. f ( x ) = 1 2 x 2 , x [0, 1]. 15. f ( x ) = 1 2 x x 2 1 1 2 ln ( x + x 2 1), x [1, 2]. 16. f ( x ) = cosh x , x [0, ln 2]. 17. f ( x ) = 1 2 x 3 x 2 + 3 2 sin 1 ( 1 3 3 x ), x [0, 1]. 18. f ( x ) = ln ( sin x ), x [ 1 6 π , 1 2 π ]. The equations in Exercises 19–24 give the position of a particle at each time t during the time interval speci±ed. Find the ini- tial speed of the particle, the terminal speed, and the distance traveled. 19. x ( t ) = t 2 , y ( t ) = 2 t , from t = 0t o t = 3. 20. x ( t ) = t 1, y ( t ) = 1 2 t 2 , from t = o t = 1. 21. x ( t ) = t 2 , y ( t ) = t 3 , from t = o t = 1. 22. x ( t ) = a cos 3 t , y ( t ) = a sin 3 t , from t = o t = 1 2 π . 23. x ( t ) = e t sin t , y ( t ) = e t cos t , from t = o t = π . 24. x ( t ) = cos t + t sin t , y ( t ) = sin t t cos t , from t = 0to t = π . 25. Let a > 0. Find the length of the path traced out by x ( θ ) = a ( θ sin θ ), y ( θ ) = a (1 cos θ ), as θ ranges from 0 to 2 π . 26. Let a > 0. Find the length of the path traced out by x ( θ ) = 2 a cos θ a cos 2 θ , y ( θ ) = 2 a sin θ a sin 2 θ , as θ ranges from 0 to 2 π 27. (a) Let a > 0. Find the length of the path traced out by x ( θ ) = 3 a cos θ + a cos 3 θ , y ( θ ) =

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SalasSV_09_08_ex - 9.8 ARC LENGTH AND SPEED 573 EXERCISES...

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