SalasSV_09_05

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Unformatted text preview: ) = π + 3 3 ∼ 8. 34. = Remark We could have done Example 3 more efficiently by exploiting the symmetry of the region. The region is symmetric about the x-axis. Therefore A=2 π π/3 1 [1 2 − 2 cos θ ]2 d θ − 2 0 π /3 1 [1 2 − 2 cos θ ]2 d θ . 9.5 AREA IN POLAR COORDINATES 551 common to the circle r = 2 sin θ and the limaçon r = − sin θ is indicated in Figure 9.5.7. The θ coordinates of the points of intersection can be found by solving the two equations simultaneously: Example 4 The region 3 2 θ= π 2 2 sin θ = Thus, the area of area of = 0 3 2 − sin θ , sin θ = 1 , 2 and θ = 1π, 5π. 6 6 θ= π 5π/6 1 [2 sin θ ]2 2 5π 6 r = 2 sinθ can be represented as follows: π /6 1 [2 sin θ ]2 d θ 2 [1, ] π 2 θ= π 6 + 5π/6 π/6 13 [ 22 − sin θ ]2 d θ + dθ ; Ω polar axis or, by the symmetry of the region, r= area of =2 0 π /6 1 [2 sin θ ]2 d θ 2 +2 π /2 π/6 3 2 – sin θ 13 [ 22 − sin θ ]2 d θ . As you can verify, the area of EXERCISES 9.5 is 5 π − 4 15 8 √ 3 ∼ 0. 68. = Figure 9.5.7 Calculate the area enclosed by the given curve. Take a > 0. 1. r = a cos θ from θ = −1π 2 −1π 6 to θ = 1 π. 2 18. Outside r = 1 − cos θ , but inside r = 1 + cos θ . 19. Inside r = 4, and to the right of r = 2 sec θ . 20. Inside r = 2, but outside r = 4 cos θ . 21. Inside r = 4, and between the lines θ = 1 π and r = 2 sec θ . 2 22. Inside the inner loop of r = 1 − 2 sin θ . 23. Inside one petal of r = 2 sin 3θ . 24. Outside r = 1 + cos θ , but inside r = 2 − cos θ . 25. Interior to both r = 1 − sin θ and r = sin θ . 26. Inside one petal of r = 5 cos 6θ . 2. r = a cos 3θ from θ = to θ = 1 π . 6 √ 1 3. r = a cos 2θ from θ = − 4 π to θ = 1 π . 4 4. r = a(1 + cos 3θ ) 5. r 2 = a2 sin2 θ . 7. r = tan 2θ 8. r = cos θ , 9. r = 2 cos θ , from θ = − 1 π to θ = 1 π . 3 3 6. r 2 = a2 sin2 2θ . θ = 0, and the rays and the rays θ= 1 π. 8 Calculate the area of the given region. and the rays r = sin θ , r cos...
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This note was uploaded on 10/12/2010 for the course MATH 12345 taught by Professor Smith during the Spring '10 term at University of Houston - Downtown.

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