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Unformatted text preview: PRACTICE MIDTERM If some of these problems have multiple parts, you should be happy because otherwise they would have been 1 problem that doesnt walk you through the steps to solve it. Step 1: Please understand exactly what the problem means . Maybe even sketch some things. Step 2: Figure out what you need to find, and the calculations you need to set up to find them. Step 3: Do the calculations. And youre done with the problem. 1. Sketch the region of integration for 1 1 2 y xy dxdy e x . Evaluate the integral, but please do it by first reversing the order of integration. A: 1 1 2 y xy dxdy e x = 1 0 0 2 x xy dydx e x = 1 2 dx xe x = 1 2 1 du e u = 2 1- e . 2. a) Find the area of the region bounded by the curve x 2 + y 2 = 1, to the right of x = 0 and above the line y = 0. b) Now, find the center of this region, by averaging: integrate x over the entire region and divide by the area found in a). Similarly for y . Put the results together into a vector. Thus you have found the average of the position vector ( x , y ). Realize why. A: a) 4 2 2 cos 1 cos 1 2 / 2 / 2 1 2 = + = =- dt t dt t dx x But then, we already knew this was a quarter of the unit disk....
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- Spring '08