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Unformatted text preview: MATH 251-3, Fall 2007 Simon Fraser University Assignment 9: Solutions Additional Question Due: 4:30pm, Monday 26 November 2007 1. Sketch the region of integration, and evaluate the following integral: Z 4 Z 2 √ y √ y cos( x 4 ) + x sin y dxdy Solution: The region of integration is the region above the x-axis and below the parabola x = √ y , that is, y = x 2 , for x between 0 and 2 (thus y is between 0 and 4). We are unable to integrate cos( x 4 ) with respect to x ; it has no elementary anti-derivative. However, we can try to change the order of integration. (This is necessary for the first term in the integrand; for the x sin y term, the order of integration seems immaterial, though integrating first with respect to x , then y will require an integration by parts.) Sketching the region and changing the order, we find Z 4 Z 2 √ y √ y cos( x 4 ) + x sin y dxdy = Z 2 Z x 2 √ y cos( x 4 ) + x sin y dydx = Z 2 2 3 y 3 / 2 cos( x 4 )- x cos y x 2 y =0 dx = Z 2 2 3 x 3 cos( x 4 )...
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