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Unformatted text preview: MATH 52 MIDTERM 1 SOLUTIONS (AUTUMN 2003) 1. Evaluate the following double integrals. (a) Evaluate the double integral integraldisplay 1 integraldisplay 4- x x 2 y dy dx Solution. integraldisplay 1 integraldisplay 4- x x 2 y dy dx = integraldisplay 1 y 2 vextendsingle vextendsingle vextendsingle vextendsingle 4- x x dx = integraldisplay 1 16- 8 x dx = 16 x- 4 x 2 vextendsingle vextendsingle vextendsingle vextendsingle 1 = 12 (b) Sketch the region of integration of the integral in part (a). 1 2 1 2 3 4 R (c) What quantity does the integral represent? Solution. There are many possible answers. It could represent the volume of the solid above the region of integration and below the graph of f ( x, y ) = 2 y . It could also represent the mass of the region of integration with mass density ( x, y ) = 2 y . 2. Consider the double integral integraldisplay 3 integraldisplay 9 y 2 y cos( x 2 ) dx dy (a) Sketch the region of integration....
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