Inner loop 22 find the area enclosed by the loop of

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Chapter 1 / Exercise 1.3
College Physics
Serway/Vuille
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(inner loop) 22. Find the area enclosed by the loop of the strophoid . 23–28 Find the area of the region that lies inside the first curve and outside the second curve. 23. , 24. , 25. , 26. , , 28. , 29–34 Find the area of the region that lies inside both curves. 29. , 30. , , 32. , 33. , 34. , , , 35. Find the area inside the larger loop and outside the smaller loop of the limaçon . 36. Find the area between a large loop and the enclosed small loop of the curve . 37–42 Find all points of intersection of the given curves. 37. , 38. , 39. , 40. , , 42. , r 2 cos 2 r 2 sin 2 r sin 2 r sin 41. r sin 3 r cos 3 r 1 r 2 sin 2 r 1 sin r 1 cos r 3 sin r 1 sin r 1 2 cos 3 r 1 2 cos b 0 a 0 r b cos r a sin r 2 cos 2 r 2 sin 2 r 3 2 sin r 3 2 cos r cos 2 r sin 2 31. r 1 cos r 1 cos r sin r s 3 cos r 2 sin r 3 sin r 1 cos r 3 cos 27. r 3 sin r 2 sin r 2 r 2 8 cos 2 r 1 r 1 sin r 1 r 2 cos r 2 cos sec r 1 2 sin 21. r 2 sin 6 r 3 cos 5 1–4 Find the area of the region that is bounded by the given curve and lies in the specified sector. 1. , 2. , 3. , 4. , 5–8 Find the area of the shaded region. 5. 6. 8. 9–14 Sketch the curve and find the area that it encloses. 9. 10. 12. 13. 14. ; 15–16 Graph the curve and find the area that it encloses. 15. 16. 17–21 Find the area of the region enclosed by one loop of the curve. 17. 18. r 4 sin 3 r sin 2 r 2 sin 3 sin 9 r 1 2 sin 6 r 2 cos 2 r 2 cos 3 r 2 sin r 2 4 cos 2 11. r 3 1 cos r 3 cos r= sin   r=4+3   sin   ¨ 7. r=1+ cos   ¨ r=oe ¨ 0 r s sin 3 2 3 r sin 2 r e 2 0 4 r 2 EXERCISES 10.4
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College Physics
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Chapter 1 / Exercise 1.3
College Physics
Serway/Vuille
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49–52 Use a calculator to find the length of the curve correct to four decimal places. 49. 50. 51. 52. ; 53–54 Graph the curve and find its length. 53. 54. 55. (a) Use Formula 10.2.7 to show that the area of the surface generated by rotating the polar curve (where is continuous and ) about the polar axis is (b) Use the formula in part (a) to find the surface area gener- ated by rotating the lemniscate about the polar axis. 56. (a) Find a formula for the area of the surface generated by rotating the polar curve , (where is continuous and ), about the line . (b) Find the surface area generated by rotating the lemniscate about the line . 2 r 2 cos 2 2 0 a b f a b r f r 2 cos 2 S y b a 2 r sin r 2 dr d 2 d 0 a b f a b r f r cos 2 2 r cos 4 4 r 1 cos 3 r sin 2 r 4 sin 3 r 3 sin 2 ; 43. The points of intersection of the cardioid and the spiral loop , , can’t be found exactly. Use a graphing device to find the approximate values of at which they intersect. Then use these values to estimate the area that lies inside both curves. 44. When recording live performances, sound engineers often use a microphone with a cardioid pickup pattern because it sup- presses noise from the audience. Suppose the microphone is placed 4 m from the front of the stage (as in the figure) and the boundary of the optimal pickup region is given by the car- dioid , where is measured in meters and the microphone is at the pole. The musicians want to know the area they will have on stage within the optimal pickup range of the microphone. Answer their question.

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