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Fall 2009 Prelim 2

# Fall 2009 Prelim 2 - dz dy dx(b Evaluate the integral 4...

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MATH 1920, FALL 2009 PRELIM 2 SHOW ALL WORK. NO CALCULATORS. NAME: STUDENT ID #: SECTION #: Who is your TA? PROBLEM SCORE 1a 1b 2a 2b 2c 2d 3a 3b 4 5 6a 6b TOTAL

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1. A wire of density δ ( x ) = 2 + x lies along the parabola y = 4 - x 2 in the plane. The wire extends from ( - 1 , 3) to (2 , 0). (a) Write an integral for the mass of the wire. You need not evaluate the integral. (b) Write integrals for the coordinates of the center of mass. You need not evaluate the integrals. 2. In the plane, consider the region R that is outside the circle r = 1 and inside the circle r = 2 cos( θ ). (a) Sketch the circle r = 1 and the circle r = 2 cos( θ ). Find the intersection points. (b) Write a double integral (or a sum of double integrals) for the area of R with the order of integration dθ dr . (c) Write a double integral (or a sum of double integrals) for the area of R with the order of integration dr dθ . (d) Calculate the area of R . 3. Consider the triple integral Z 1 0 Z 1 z Z 1 - y 0 ( x - x 3 ) cos ( y 2 ) dx dy dz. (a) Change the order of integration to
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Unformatted text preview: dz dy dx . (b) Evaluate the integral. 4. Calculate the circulation of the ﬁeld F = 3 xy i + sin 16 ( y ) j clockwise around the boundary of the triangle with vertices (0 , 0), (1 , 1), and (2 , 0). 5. Consider the surface S deﬁned by x 2 + y 2 = 4, 0 ≤ z ≤ 3, and oriented with normal vectors n pointing away from the z-axis. Compute the ﬂux of the vector ﬁeld F = 2 xz i + yz 2 j + p x + y 2 k across S in the direction of n . 6. Consider a paraboloid hut with base z = 0 and roof z = 2-x 2-y 2 . We assume that ≤ a < √ 2, so that the hut is not inside the cylinder x 2 + y 2 = a 2 . Let R be the region that is inside the hut and inside the cylinder x 2 + y 2 = a 2 . (a) Compute the volume of R . (b) Set b = a 2 . Find b if the volume of R is equal to 3 4 (the volume of the hut). 1...
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Fall 2009 Prelim 2 - dz dy dx(b Evaluate the integral 4...

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