Math252Sols5F06

Math252Sols5F06 - QUIZ 5 (CHAPTER 18) SOLUTIONS MATH 252...

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QUIZ 5 (CHAPTER 18) SOLUTIONS MATH 252 – FALL 2006 – KUNIYUKI 105 POINTS TOTAL, BUT 100 POINTS = 100% Show all work, simplify as appropriate, and use “good form and procedure” (as in class). Box in your final answers! No notes or books allowed. A scientific calculator is allowed. 1) C consists of the curves C 1 and C 2 in the xy -plane. That is, C = C 1 C 2 . The curve C 1 is the directed line segment from 0,0 ( ) to 4,2 ( ) , and the curve C 2 is the portion of the parabola x = y 2 directed from ( ) to 9,3 ( ) . If the force at x , y ( ) is F x , y ( ) = 4 y 3 , 3 x , find the work done by F along C . It is recommended that you write your final answer as a decimal. (25 points) Parameterize C 1 : Parametric equations for C 1 (and dx and dy ) are given by: x = 4 t dx = 4 dt y = 2 t dy = 2 dt t :0 1 ( ) This is because the displacement vector from ( ) to ( ) is given by v = . We could also use: x = 2 t dx = 2 dt y = t dy = dt t 2 ( ) Parameterize C 2 : x = t 2 dx = 2 t dt y = t dy = dt t :2 3 ( )
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Compute the work integral along C 1 : F d r C 1 = 4 y 3 ,3 x dx , dy C 1 = 4 y 3 dx + 3 xdy Think: M dx + N dy C 1 C 1 = 4 2 t ( ) 3 4 dt ( ) + 3 4 t ( ) 2 dt ( ) 0 1 = 128 t 3 dt + 24 tdt 0 1 = 32 t 4 + 12 t 2 0 1 = 32 1 ( ) 4 + 12 1 ( ) 2 0 = 32 + 12 = 44 Compute the work integral along C 2 : The total work along C is given by: F d r C = F d r C 1 + F d r C 2 = 44 + 356.6 = 400.6 F d r C 2 = 4 y 3 x dx , dy C 2 = 4 y 3 dx + 3 C 2 = 4 t ( ) 3 2 ( ) + 3 t 2 ( ) dt ( ) 2 3 = 8 t 4 + 3 t 2 ( ) dt 2 3 = 8 t 5 5 + t 3 2 3 = 8 3 ( ) 5 5 + 3 ( ) 3 8 2 ( ) 5 5 + 2 ( ) 3 = 1944 5 + 27 256 5 + 8 = 1783 5 or 356.6
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2) Use the idea of potential functions and the Fundamental Theorem for Line Integrals to show that the following line integral is independent of path in Octant I of xyz -space and to evaluate the integral. Show all work, as we have done in class. Use good form. In particular, indicate independent variables for
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Math252Sols5F06 - QUIZ 5 (CHAPTER 18) SOLUTIONS MATH 252...

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