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# mter1 - i 1 b i 1 √ y cos(2 π y x 2 y dx B dy 4 Problem...

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Math 32B, Winter 06, Midterm 1 February 27, 2006 Name: Student #: Section: There are six problems. You have 50 minutes. Do as many problems as you can in this time, skip those which you cannot solve. Each problem is worth 5 points. You are not allowed to use calculators. For most of the problems it is a very good idea to draw a picture of the domain of integration. .......... Good luck! 1

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Problem 1 Let f be a function deFned on the rectangle R = [0 , 4] × [0 , 8]. Suppose you know that f (1 , 2) = 3, f (3 , 2) = 8, f (1 , 6) = 10, f (3 , 6) = 4. Give an approximation for the integral i i R f ( x, y ) dA by the midpoint rule Riemann sum which uses all information you have. 2
Problem 2 Calculate the following integral: i i D cos( x 2 + y 2 ) dxdy , where the domain D consists of all points in the x, y - plane whose distance to (0 , 0) is greater than or equal to r π 2 and lower than or equal to π . 3

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Problem 3 Calculate the following integral:

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Unformatted text preview: i 1 b i 1 √ y cos(2 π y x 2 ) + y dx B dy 4 Problem 4 Calculate the center of mass of a thin plate which has the shape of a triangle in the x, y-plane given by the corners (0 ,-1), (1 , 0), (0 , 1) and which has a mass density distribution given by ρ ( x, y ) = 2 x 2 . Recall that the coordinates of the center of mass are given by the Frst moments divided by the total mass. 5 Problem 5 Calculate the integral of the function i i D 3 dxdy Here the domain D is given in the new coordinates u, v by ≤ u ≤ 1 ≤ v ≤ u where u, v are deFned by the equations x = u + u 3 y = u + v + v 3 6 Problem 6 Find the surface area of the part of the paraboloid z = x 2 + y 2 that lies under the plane z = 4. The solution involves a root of 17 which you do not have to calculate. 7...
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mter1 - i 1 b i 1 √ y cos(2 π y x 2 y dx B dy 4 Problem...

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