fall17mth143.practice2.11-PolarCalc.pdf

# 1 2cos 3 θ area answers submitted incorrect correct

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1 + 2cos ( 3 θ ) . Area = Answer(s) submitted: (incorrect) Correct Answers: pi/3+sqrt(3) 6. (1 point) Find the area of the shaded region, where the polar curve is given by r = θ . You may need to click on the figure to obtain a clear view of the picture. Area = Answer(s) submitted: (incorrect) Correct Answers: 1/6*piˆ3 7. (1 point) Find the area lying outside r = 4cos θ and inside r = 2 + 2cos θ . Area = Solution: SOLUTION The graph of r = 4cos θ is a circle with radius 2 centered on (2,0), while r = 2 + 2cos θ is a cardioid that entirely encloses the circle. Therefore, to find the area, we subtract the area enclosed by the circle from that enclosed by the cardioid. The cardioid, r = 2 + 2cos θ , is traced out as θ goes from θ = 0 to θ = 2 π . Thus, remembering that the area enclosed by a region lying within the polar equation r = f ( θ ) is Area = R β α 1 2 ( f ( θ )) 2 d θ , Area of cardioid = Z 2 π 0 1 2 ( 2 + 2cos θ ) 2 d θ . The circle is traced out as θ goes from θ = 0 to θ = π , so its area is Area of circle = Z π 0 1 2 ( 4cos θ ) 2 d θ . Of course, because the circle has radius 2, we know this second integral will evaluate to 4 π . We can find the area of the cardioid, Z 2 π 0 1 2 ( 2 + 2cos θ ) 2 d θ = 1 2 Z 2 π 0 4 ( 1 + 2cos θ + cos 2 θ ) d θ , numerically by using a calculator, or by parts, or using a table of integrals, to find that the area of the cardioid is A = 6 π . Thus, the area enclosed by the cardioid that lies outside of the circle is given by A = 6 π - 4 π = 2 π . Answer(s) submitted: (incorrect) Correct Answers: pi*2*2/2 8. (1 point) Find the exact length of the polar curve r = cos 2 ( θ / 2 ) .

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