MAT-252 Sample Quiz 4 Solutions SP17 - MAT-252 Sample Quiz 4 Solutions 1 Calculate the shaded area in the figure below using the indicated methods y

MAT-252 Sample Quiz 4 Solutions SP17 - MAT-252 Sample...

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MAT-252 Sample Quiz 4 Solutions 1) Calculate the shaded area in the figure below using the indicated methods. a) Integrating with respect to x . Break into two integrals. Find the area under the sine function on the interval 1 , 0 and add the area under the linear function on the interval 2 , 1 . 2 1 2 1 0 2 1 1 0 2 1 2 2 cos 2 2 2 sin x x x dx x dx x A     2 3 2 2 1 2 1 2 2 0 cos cos 3 2 2 2 1 1 1 1 0 2 2       A 137 . 1 2 1 2 A b) Integrating with respect to y . dy y y dy x x dA left right 1 sin 2 2 slice horiz. of area 1 0 1 sin 2 2 dy y y A The last term of the integral requires integration by parts and won’t be easy , so just use your calculator to confirm the answer from part (a). 137 . 1 A y x x y right 2 2 y x y x x y left 1 sin 2 2 sin 2 (1, 1) First note that the graphs intersect at the point (1, 1). Linear Function: 1 x 1 1 2 y Sine Function: 1 x 1 2 sin y Also , both functions cross the x -axis when 2 x . Linear Function: 2 x 0 2 2 y Sine Function: 2 x   0 sin y dy x right x left