old quiz 4 2008

old quiz 4 2008 - Math 2401 Quiz #4 April 17Tom Morley 8:00...

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Unformatted text preview: Math 2401 Quiz #4 April 17Tom Morley 8:00 am Problem 1 (10 points) George Mallory needs to know the following : Let S be the surface of the hyperboloid of two sheets x^2 y^2 z^2 4 above the plane z 3 and below the plane z 5. Set up the double integral to find the surface area. In[65]:= In[66]:= Out[66]= Clear x, y, z x r Cos Θ r Cos Θ y r Sin Θ In[67]:= Out[67]= In[68]:= Out[68]= In[69]:= Out[69]= In[70]:= Out[70]= In[71]:= Out[71]= In[72]:= Out[72]= In[73]:= r Sin Θ z 4 4 r2 x, y, z r2 r^2 pos r Cos Θ , r Sin Θ , 4 D pos, r Cos Θ , Sin Θ , 2 r D pos, Θ r Sin Θ , r Cos Θ , 0 Cross D pos, r , D pos, Θ 2 r2 Cos Θ , Simplify Sqrt r2 Solve r Solve r 4 r4 r^2 3^2 5 4, r 21 4, r 2 r2 Sin Θ , r . Simplify Out[73]= In[74]:= Out[74]= 5,r r^2 5^2 In[75]:= Out[75]= 21 , r 2 quiz4ans.nb 2Π 21 In[76]:= 0 5 r2 4 r4 rΘ 7 Out[76]= 21 2 85 6 85 Π Problem 2 (10 points) Find the value of the line integral C 2 x y Sin x2 y dx x2 Sin x2 y dy a) Where C is the curve from (1,1) to (4,8) given by: y = t^3, x = t^2 b) Where C is the curve from (1,1) to (4,8) given by: Clear x, y, g g x ,y Cos x2 y D g x, y , x , D g x, y , y 2 x y Sin x2 y , x2 Sin x2 y g 4, 8 Cos 1 g 1, 1 Cos 128 Cos x ^ 2 y Problem 3 (10 points) Using Green’s Theorem, convert the line integral: double integral. P xy Q x2 y2 xy C x y dx x2 y2 dy , where C is the curve x^2 + (y/2)^2 = 1 (an ellipse) to a x2 y2 integrand x 2 x y2 D Q, x D P, y quiz4ans.nb 3 Solve x ^ 2 y 1 1 2 2 y 2 ^2 1 x2 ,y 1, y 2 1 x2 2 1 x2 x 1 x2 2 x y2 y x Integrate 0 Integrate x 2 x y2 , y, 2 1 x2 , 2 1 x2 , x, 1, 1 ...
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