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e-reader-frontend.macmillanlearning.com/?ro=macmillanlearning.com?rid=7575ee99b808004-04aa4216-4216867-867e-e99e00b1010f6868e33?req_id=1313f486486c22619-619b429-429e... All Bookmarks 1414. Find the area of the shaded region in Figure 1818 enclosed by the circle r=12r=\frac{1}{2} and a petal of the curve r=cos3θr=\cos 3\theta. Hint: Compute the area of both the petal and the region inside the petal and outside the circle. Rogawski et al., Calculus: Early Transcendentals, 44e, (C) 20192019 W. H. Freeman and Company FIGURE 1818 1515. Find the area of the inner loop of the lima\c{c}on with polar equation r=2cosθ1r=2\cos \theta-1 (Figure 1919). 1616. Find the area of the shaded region in Figure 1919 between the inner and outer loop of the lima\c{c}on r=2cosθ1r=2\cos \theta-1.

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Q. e-reader-frontend.macmillanlearning.com/?ro=macmillanlearning.com?rid=7575ee99b808004-04aa4216-4216867-867e-e99e00b1010f6868e33?req_id=1313f486486c22619-619b429-429e... All Bookmarks 1414. Find the area of the shaded region in Figure 1818 enclosed by the circle r=12r=\frac{1}{2} and a petal of the curve r=cos3θr=\cos 3\theta. Hint: Compute the area of both the petal and the region inside the petal and outside the circle. Rogawski et al., Calculus: Early Transcendentals, 44e, (C) 20192019 W. H. Freeman and Company FIGURE 1818 1515. Find the area of the inner loop of the lima\c{c}on with polar equation r=2cosθ1r=2\cos \theta-1 (Figure 1919). 1616. Find the area of the shaded region in Figure 1919 between the inner and outer loop of the lima\c{c}on r=2cosθ1r=2\cos \theta-1.
  1. Calculate Circle Area: Calculate the area of the circle with radius r=12r = \frac{1}{2}.\newlineArea of a circle = πr2\pi r^2\newlineCalculation: π(12)2=π4\pi \cdot (\frac{1}{2})^2 = \frac{\pi}{4} square units.
  2. Calculate Petal Area: Calculate the area of one petal of the curve r=cos3θr = \cos 3\theta.\newlineArea of one petal = 12\frac{1}{2} (r2)dθ\int (r^2) d\theta from 00 to π3\frac{\pi}{3} (since the petal repeats every π3\frac{\pi}{3}).\newlineCalculation: 12\frac{1}{2} * (cos3θ)2dθ\int (\cos 3\theta)^2 d\theta from 00 to π3\frac{\pi}{3}.\newlineUsing the double angle identity, 12\frac{1}{2}00.\newlineArea = 12\frac{1}{2} * 12\frac{1}{2}22 from 00 to π3\frac{\pi}{3} = 12\frac{1}{2}55 square units.
  3. Calculate Shaded Region Area: Calculate the area of the region inside the petal and outside the circle.\newlineArea of shaded region = Area of petal - Area of circle.\newlineCalculation: π12π4=π6\frac{\pi}{12} - \frac{\pi}{4} = -\frac{\pi}{6} square units.

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