q4-sol-1042-sp08

q4-sol-1042-sp08 - Calculus II-1042 Solution of Quiz 4 1....

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Calculus II—1042 Solution of Quiz 4 1. Given are graphs of y = sin x, and y = cos x . Find the area bounded by y = sin x , y = cos x , and 0 x π . Solution: As we notice it is a vertically simple region, and the point of intersection between 0 x π , where sin x = cos x is π/ 4 y x 0 1 /4 π sin x sin x cos x cos x π Area = Area A 1 +Area A 2 A 1 = R x large x small ( y upper - y lower ) dx = Z π/ 4 0 (cos x - sin x ) dx = sin x + cos x | π/ 4 0 = sin( π/ 4) + cos( π/ 4) - (sin 0 + cos 0) = 2 2 + 2 2 - (0 + 1) = 2 - 1 A 2 = Z x large x small ( y upper - y lower ) dx = Z π π/ 4 (sin x - cos x ) dx = - cos x - sin x | π π/ 4 = - cos( π ) - sin( π ) - ( - cos( π/ 4) - sin( π/ 4)) = ( - ( - 1) - 0) - ˆ - 2 2 - 2 2 ! = 1 + 2 Area of the shaded region is A 1 + A 2 = 2 - 1 + 1 + 2 = 2 2 sq. units 2. Given are graphs of x + y 2 = 0 , and x + 3 y 2 = 2. Find the area bounded by x + y 2 = 0, and x + 3 y 2 = 2. Solution:
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q4-sol-1042-sp08 - Calculus II-1042 Solution of Quiz 4 1....

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