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**Unformatted text preview: **θ , θ = 0, θ = 0, θ = 1π. 4 θ = 1π. 4 θ = 1π. 4 27. Outside r = cos 2θ , but inside r = 1. 28. Interior to both r = 2a cos θ and r = 2a sin θ , a > 0. 29. Find the area of the region that is common to the three circles: r = 1, r = 2 cos θ , and r = 2 sin θ . 30. Find the area of the region outside the circle r = a and inside the lemniscate r 2 = 2a2 cos 2θ . 31. Fix a > 0 and let n be a positive integer. Prove that the petal curves r = a cos 2nθ and r = a sin 2nθ all enclose exactly the same area. Find the area. 32. Fix a > 0 and let n be a positive integer. Prove that the petal curves r = a cos ([2n + 1]θ ) and r = a sin ([2n + 1]θ all enclose exactly the same area. Find the area. Centroids in Polar Coordinates Let be the region bounded by the polar curve r = ρ (θ ) between θ = α and θ = β . Since the centroid of a triangle lies on each median, two-thirds of the distance from the vertex to the opposite side (see Exercise 29, Section 6.4), it follows that the x and y coordinates of the centroid of the 10. r = 1 + cos θ , r = cos θ , and the rays θ = 0, θ = 1 π . 2 11. r = a(4 cos θ − sec θ ) 12. r =
1 2 and the rays θ = 0, sec2 1 θ 2 and the vertical line through the origin. Find the area between the curves. 13. r = eθ , the rays 14. r = eθ , the rays 15. r = eθ , the rays 16. r = eθ , the rays 0 ≤ θ ≤ π; r = θ, 0 ≤ θ ≤ π; θ = 0, θ = π . 2π ≤ θ ≤ 3π ; r = θ , 0 ≤ θ ≤ π ; θ = 0, θ = π . 0 ≤ θ ≤ π ; r = eθ/2 , 0 ≤ θ ≤ π ; θ = 2π , θ = 3π . 0 ≤ θ ≤ π ; r = eθ , 2π ≤ θ ≤ 3π ; θ = 0, θ = π . Represent the area by one or more integrals. 17. Outside r = 2, but inside r = 4 sin θ . 552 CHAPTER 9 THE CONIC SECTIONS; POLAR COORDINATES; PARAMETRIC EQUATIONS “triangular” region shown in the ﬁgure are given approximately by x=
2 ρ (θ ) cos θ 3 36. The region enclosed by r = 2 + sin θ . y = 2 ρ (θ ) sin θ 3 c In Exercises 37 and 38 ,use a graphing utility to draw the polar curve. Then use a CAS to ﬁnd the area of the region it...

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