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**Unformatted text preview: **encloses.
37. r = 2 + cos θ . 38. r = 2 cos 3θ . These approximations improve as the triangle narrows.
θ =β
y c In Exercises 39 and 40, use a graphing utility to drawn the polar curves. Then use a CAS to ﬁnd the area of the region inside the ﬁrst curve and outside the second curve.
39. r = 4 cos 3θ , r = 2. 40. r = 2 cos θ , r = 1 − cos θ . c 41. The curve whose equation in rectangular coordinates is
x ρ (θ )
y θ Ω x r = ρ (θ ) y2 = x2 a−x , a+x a > 0, is called a strophoid. (a) Show that the polar equation of this curve has the form r = a cos 2θ sec θ . (b) Use a graphing utility to draw the graph of the curves for a = 1, 2, and 4. (c) Let a = 2. Find the area inside the loop. θ =α 33. Following the approach used in Section 6.4, show that the rectangular coordinates of the centroid of are given by xA =
1 3 β α [ρ (θ )]3 cos θ d θ , . yA = 1 3 β α c 42. The curve whose equation in rectangular coordinates is
[ρ (θ )]3 sin θ d θ (x2 + y2 )2 = ax2 y, a > 0, where A is the area of is called a bifolium. (a) Show that the polar equation of this curve has the form r = a sin θ cos2 θ . (b) Use a graphing utility to draw the graph of the curves for a = 1, 2 and 4. (c) Let a = 2. Find the area inside one of the loops. In Exercises 34–36, use the result of E...

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