Prelim 3 Sp'06 - Math 192 Prelim 3 Calculators are not...

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Math 192 Prelim 3, April 27, 2006 Calculators are not allowed. Write your section number and TA’s name on the front of your workbook. This exam is worth 100 points. Point values for each problem are in parentheses. Show all your work . 1. (15) For each of the following, set up an integral using the indicated coordinates to find the volume but do not evaluate the integral. i) Cylindrical coordinates: The region that lies inside the sphere x 2 + y 2 + z 2 = 2 and outside the cylinder x 2 + y 2 = 1. ii) Spherical coordinates: The region enclosed by the cone z = x 2 + y 2 between the planes z = 1 and z = 2 . . 2. (15) Find the mass of a wire that lies along the curve r ( t ) = ( t 2 - 1) j + 2 tk, 0 t 1, if the density at any point on the curve is given by δ = 3 t/ 2. 3. (15) Find the flow of the vector field F ( x, y, z ) = 6 zi + y 2 j + 12 xk along the curve C given by r ( t ) = (sin t ) i + (cos t ) j + ( t/ 6) k, 0 t π
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