Chapter 9 Quiz Solutions

This occurs when 23 the curve is symmetric about the

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Unformatted text preview: op meets the outside loop when r = 0, that is, when −1/2 = cos(θ). This occurs when θ = ±2π/3. The curve is symmetric about the x-axis so we can just compute the area of the part that’s above the x-axis and double it. (area between loops) = (area inside big loop) − (area inside small loop) 2π/3 = 2 ∗ (1/2) π (1/2 + cos(θ))2 dθ − 2 ∗ (1/2) 0 (1/2 + cos(θ))2 dθ 2π/3 Using the double angle formula we find that (1/2 + cos(θ))2 dθ = 1/4 + cos(θ) + cos2 (θ)dθ 1 + cos(2θ) dθ 2 = 1/4 + cos(θ) + = 3/4 + cos(θ) + (1/2) cos(2θ)dθ =(3/4)θ + sin(θ) + (1/4) sin(2θ) + C so the area inside the region is 2π/3 π ((3/4)θ + sin(θ) + (1/4) sin(2θ)|0 − ((3/4)θ + sin(θ) + (1/4) sin(2θ)|2π/3 √ = (1/4)(3 3 + π ). Π 7Π 2Π 2 12 1.5 5Π 12 3 3 3Π Π Π 4 4 1.0 5Π Π 6 6 11 Π Π 0.5 12 12 Π 1.5 1.0 0.5 0.5 1.0 1.5 11 Π 0 Π 0.5 12 12 5Π Π 6 6 1.0 3Π 4 Π 2Π 3 Π 7Π 12 1.5 Π 5Π 2 12 Page 2 3 4...
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