Suppose c is a smooth simple closed curve in the

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(2) Suppose C is a smooth simple closed curve in the plane which bounds a region R . Which of the following path integrals must always equal twice the area of R ? (The path integral is taken counterclockwise around C .) (A) I xdy + ydx (B) I xdy - ydx (C) I ydy - xdx (D) I ydy + xdx (E) 2 I ydy - xdx (F) None of the above. Answer to 2 :
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4 (3) Which of the following vector fields F : R 2 R 2 in the plane are conservative, in the sense that the value of a path integral of the vector field always just depends on the endpoints of the path? The formulas below express the value of F at a given point ( x, y ) as a linear combination of i = (1 , 0) and j = (0 , 1) (I) F = x i + y j . (II) F = y i + x j . (III) F = y 2 i + x 2 j . Answer to 3:
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5 (4) Find the area of the parallelogram in three space whose vertices are at (0 , 0 , 0), (1 , 1 , 1), (1 , - 1 , 0), (2 , 0 , 1). Answer to 4:
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6 (5) A silo is to be built having a flat circular bottom, cylindrical side, and a hemispher- ical top. The silo is to have total volume (including the top cap) V = 900 π cubic feet. If the cost of the metal is 5 dollars/sq.ft. for the roof, 2 dollars/sq.ft. for the sides, and 1 dollar/sq.ft. for the floor, what should the ratio of height ( h ) to radius ( r ) of the cylindrical portion of the silo be in order to minimize the total cost of the materials needed? You can use without proof the usual formulas for the volume and surface areas of spheres and cylinders. For example, a sphere of radius r has volume 4 πr 3 / 3 and surface area 4 πr 2 . The volume of a solid cylinder of radius r and height h is πr 2 h .
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