Explanation Given: The limacon r = 1 + 2 cos θ Concept Used: Area of a region bounded by r 1 ( θ ) ≤ r ≤ r 2 ( θ ) and θ 1 ≤ θ ≤ θ 2 is given as: ∫ θ 1 θ 2 ∫ r 1 ( θ ) r 2 ( θ ) r d r d θ Calculation: To find the area inside the inner loop of limacon r = 1 + 2 cos θ , first define the region. Note that the given region is inner loop of limacon r = 1 + 2 cos θ shown below by shaded area: Now, to find the range of angle find the values where curve goes through origin, which can be found by setting the equation of limscon equals to 0 and solving, as 1 + 2 cos θ = 0 cos θ = − 1 2 θ = 2 π 3 , 4 π 3 Thus, θ varies from 2 π 3 to 4 π 3 and r varies from 0 to 1 + 2 cos θ So, the limit of integration in polar coordinates would be 0 ≤ r ≤ 1 + 2 cos θ , and 2 π 3 ≤ θ ≤ 4 π 3 Thus, the area of given shaded region is calculated as ∫ 2 π 3 4 π 3 ∫ 0 1 + 2 cos θ r d r d θ = ∫ 2 π 3 4 π 3 [ ∫ 0 1 + 2 cos θ r d r ] d θ = ∫ 2 π 3 4 π 3 [ r 2 2 ] 0 1 + 2 cos θ d θ = 1 2 ∫ 2 π 3 4 π 3 [ ( 1 + 2 cos θ ) 2 − 0 ] d θ = 1 2 ∫ 2 π 3 4 π 3 ( 1 + 4 c o s θ + 4 cos 2 θ ) d θ = 1 2 ∫ 2 π 3 4 π 3 d θ + 1 2 ∫ 2 π 3 4 π 3 4 cos θ d θ + 1 2 ∫<

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ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 12.3, Problem 23PS

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To find the area inside the inner loop of limacon r=1+2cosθ

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Chapter 12.3, Problem 22PS

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To find the area inside the inner loop of limacon r = 1 + 2 cos θ

Solution Summary: The author explains that to find the area inside the inner loop of limacon r=1+2mathrmcostheta , first define the region.

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To find the area inside the inner loop of limacon r=1+2cosθ

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