t2_s2_sln - YORK UNIVERSITY Faculty of Pure and Applied...

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YORK UNIVERSITY Faculty of Pure and Applied Science AS/SC/MATH 1014 3.0 M June - August 2003 Term Test 2 SOLUTIONS 1. (a) (5 points) Find all points of intersection of r = 3sin θ and r = 3 - 3sin θ. Answer: Solving the equations simultaneously, 3sin θ = 3 - 3sin θ 6sin θ = 3 sin θ = 1 2 θ = π 6 or 5 π 6 . Since 3sin π 6 = 3 - 3sin 5 π 6 = 3 2 , we obtain the following two points of intersection P 1 ( 3 2 , π 6 ) and P 2 ( 3 2 , 5 π 6 ) . Both equations r = 3sin θ and r = 3 - 3sin θ remain unchanged when we replace ( r,θ ) by ( - r, - θ ) , and their graphs represent a symmetric about the line θ = π 2 circle and cardioid, respectively. From the graphs, we obtain that the pole is the third point of intersection. (b) (7 points) Find the total area of the region enclosed by the graph of the function r = 2cos2 θ. Answer: The equation r = 2cos2 θ has the form r = a cos nθ, where a = 2 , and n = 2 is even. So, its graph is a four-leaved rose with the length of a leaf equals 2 . Replacements θ by - θ, and θ by π - θ produce an equivalent equation, so the graph is symmetric about the polar axis, the line
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This note was uploaded on 05/18/2010 for the course MATH 1014 taught by Professor Ganong during the Spring '09 term at York University.

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t2_s2_sln - YORK UNIVERSITY Faculty of Pure and Applied...

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