Quiz6-3_22 - Name: You have fifteen minutes to complete...

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Unformatted text preview: Name: You have fifteen minutes to complete the quiz. If anything seems unclear, please ask. 1. Evaulate the integral f (x, y )dA, R where f (x, y ) = 6x2 y + 2y and R = [−1, 1] × [2, 3]. (3 points) 2. Let f (x, y ) be a general function of two variables. Sketch the region of integration, and switch the order for the integral below. (4 points) √ 2 x f (x, y )dydx 1 0 3. Set up (do not try to solve) an integral for the volume of the region where 0 ≤ x ≤ 1 − y , 0 ≤ y and 0 ≤ z ≤ x2 + ex | cos(y )|. (3 points) 1 Solutions If you have any questions and / or think you’ve spotted a mistake / typo, please get in touch with me. 1. The integral we want to solve is 3 1 6x2 y + 2ydxdy. 2 −1 The inner integral is 1 6x2 y + 2ydx = 2x3 y + 2yx −1 x=1 x=−1 = 2y + 2y − (−2y − 2y ) = 8y, and the outer integral is 3 8ydy = [4y 2 ]3 = 36 − 16 = 20. 2 2 2. I can’t get the computer to draw the region of integration nicely (sorry), √ but it’s the region between the lines y = 0 and y = x in the vertical direction, and between the lines x = 1 and x = 2 in the horizontal direction. The integral with the order switched is 1 √ 2 2 2 f (x, y )dxdy + 0 f (x, y )dxdy. 1 1 y2 3. The desired integral is 1−x 1 x2 + ex | cos(y )|dydx 0 0 (drawing a picture of the region you are integrating over might help see why this is). 2 ...
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Quiz6-3_22 - Name: You have fifteen minutes to complete...

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