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finalpracticeprobs - Practice Problems for Math 212 Final...

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Practice Problems for Math 212 Final April 19, 2006 Instructions: Below are many practice problems for the ﬁnal exam in MATH 212. You should know how to do all of these problems, but this list may not comprise all of the material to be tested on the ﬁnal exam. Note: Two points will be deducted each time you fail to quote (i.e. indicate you are using) a major theorem. For example, if you are using Green’s theorem , you must indicate it: ± C P dx + Q dy = GT ± ± D ∂Q ∂x - ∂P ∂y dx dy . 1. Evaluate the integral ± C (2 x 3 - y 3 ) dx + ( x 3 + y 3 ) dy where C is the unit circle oriented positively. 2. Find the area bounded by the x -axis and one arc of the cycloid, which is parameterized by x = a ( θ - sin θ ) y = a (1 - cos θ ) for some a > 0 and when θ ranges from 0 to 2 π . a p 2 a p x a 2 a y 1

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3. Evaluate ± ± S curl F · d S , where F = 1 3 ( - x, - 1 2 y, x y ) , and S is the union of two pieces S 1 and S 2 . S 1 is the half ellipsoid x 2 + ( y 2 ) 2 + ( z - 2 3 ) 2 = 1 and z 2. S 1 is the elliptical cylinder x 2 + ( y 2 ) 2 = 1 and 0 z 2. 4. Evaluate ± ± S curl F · d S , where F = ( x - z cos( yz ) , - y cos( yz ) , e z (1 + x 2 )), and S is the chocolate shell of an ice cream cone that has been completely dipped in chocolate fudge (i.e. every portion of the ice cream cone has been covered). 5. Is F = (cos( xy ) - xy sin( xy ) , - x 2 sin( xy )) a gradient vector ﬁeld? If so, ﬁnd a scalar potential for F . If not, say why not. 6. Evaluate
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finalpracticeprobs - Practice Problems for Math 212 Final...

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