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Fall 2007 - Enright's Class - Exam 1

Fall 2007 - Enright's Class - Exam 1 - Midterm Solutions...

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Midterm Solutions MTH 20E Note that there were three separate versions of the quiz; while the numbers may differ slightly, the techniques used in each problem remain unchanged. Question 1. The surface S has a constant normal vector; namely, A × B . Then the flux of the vector field is given by Z Z S F · d S = Z Z S F · n dV = Z Z S F · A × B k A × B dV = F · A × B k A × B k Z Z S dV = F · A × B k A × B k ( k A × B k ) = F · ( A × B ) = 6 . Question 2. By a direct application of the Change of Variables Theorem, this integral becomes Z Z Φ( D ) x dx dy = Z Z D 2 s + 3 t ( x, y ) ( s, t ) ds dt = 5 Z 1 0 Z 1 0 2 s + 3 t ds dt = 25 2 . Question 3. The corollary to Green’s Theorem states that the area of a region D is given by 1 2 R C x dy - y dx , where C is a correctly oriented closed simple curve which bounds D In this case, our region is bounded by two curves; C 1 , which is given, and C 2 := ( t, 0) , 0 t 2 πR , the curve along the x -axis from the origin to (2 πR, 0). Our parametric equation for C 1 as
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