111_quiz8_spr08_key

111_quiz8_spr08_key - x = 2 t-t 2 , y = √ t , and the...

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APMA 111 Quiz 8 March 25, 2008 NAME: SECTION: #2 (11 am) #6 (10 am) On my honor as a student, I have neither given nor received aid on this quiz. SIGNATURE: 1. [7 pts] Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases: x = 2sin t , y = 1 + cos t , 0 t 3 π/ 2. Solution: Since sin t = x/ 2 and cos t = y - 1, we can eliminate the parameter t : ( x/ 2) 2 + ( y - 1) 2 = sin 2 t + cos 2 t = 1 x 2 4 + ( y - 1) 2 = 1 . This tells us that the curve follows an ellipse. The path begins at ( x (0) ,y (0)) = (0 , 2) and ends at ( x (3 π/ 2) ,y (3 π/ 2)) = ( - 2 , 1). Along the way, it passes through the point ( x ( π ) ,y ( π )) = (0 , 0), so we know that the curve is traveled in the clockwise direction. t 1 2 3 4 x K 2 K 1 0 1 2 x as a function of t t 0 1 2 3 4 y 0 0.5 1.0 1.5 2.0 y as a function of t x K 2 K 1 0 1 2 y 0.5 1.0 1.5 2.0 parametric plot of x and y as functions of t
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Quiz 8, Page 2 of 2 March 25, 2008 2. [8 pts] Find the area enclosed by the parametric curve
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Unformatted text preview: x = 2 t-t 2 , y = √ t , and the y-axis. Solution: In order to find the points of intersection, set x = 0: 2 t-t 2 = 0 ⇒ t (2-t ) = ⇒ t = 0 , 2 and y = 0 , √ 2. Since x = 2 t-t 2 ≤ 0 for 0 ≤ t ≤ 2, AREA = Z √ 2 xdy = Z 2 ( 2 t-t 2 ) ± 1 2 √ t dt ¶ = 1 2 Z 2 ( 2 t-t 2 ) t-1 / 2 dt = 1 2 Z 2 ‡ 2 t 1 / 2-t 3 / 2 · dt = 1 2 ± 2 · 2 3 t 3 / 2-2 5 t 5 / 2 ¶fl fl fl fl 2 = 1 2 ± 4 3 2 3 / 2-2 5 2 5 / 2 ¶ = 4 √ 2 ± 1 3-1 5 ¶ = 8 √ 2 15 . t 0.5 1.0 1.5 2.0 x 0.2 0.4 0.6 0.8 1.0 x as a function of t t 0.5 1.0 1.5 2.0 y 0.2 0.4 0.6 0.8 1.0 1.2 1.4 y as a function of t x 0.2 0.4 0.6 0.8 1.0 y 0.2 0.4 0.6 0.8 1.0 1.2 1.4 parametric plot of x and y as functions of t...
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This note was uploaded on 09/21/2009 for the course MATH 1234 taught by Professor Egcdd during the Spring '09 term at Aarhus Universitet.

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111_quiz8_spr08_key - x = 2 t-t 2 , y = √ t , and the...

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