MAC2313_Test4_Spring2010

MAC2313_Test4_Spring2010 - F ). Page 3 of 4 6. (10 points...

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Page 1 of 4 MAC 2313 Name_________________________________ Test 4 Date: April 15, 2010 For full credit, clearly show all work that justifies your answer. 1. (8 points) Evaluate the iterated integral ∫∫∫ 1 0 2 0 8 x x y dzdydx xyz . 2. (8 points) Use cylindrical coordinates to set up, but do not evaluate , a triple integral that represents the volume of the solid in R 3 bounded by the cylinder 9 2 2 = + z x and the planes y = 0 and y = 4. 3. (8 points) Suppose E is the region in R 3 that lies between the sphere of radius 2 and the sphere of radius 5 in the octant in which only the x -coordinate is positive. Describe E using spherical coordinates.
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Page 2 of 4 4. (8 points each) Given the transformation v x = , ) 1 ( 2 v u y + = : (a) Graph the image under the transformation of the line segment in the uv -plane from (1,0) to (1,1). Label the endpoints of the image in the xy -plane. (b) Find the Jacobian of the transformation. 5. (5 points each) Given the vector field F ( x , y , z ) = < e x , e xy , e xyz >: (a) Compute curl( F ). (b) Compute div(
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Unformatted text preview: F ). Page 3 of 4 6. (10 points each) Evaluate each of the following line integrals: (a) ds z x C + ) 9 2 ( where C is parameterized as follows: t x = , 2 t y = , 3 t z = , 1 t . (b) r F d C where F ( x, y, z ) = xy i + yz j + zx k and C is the same curve with the same parameterization as the one in part (a); i.e. t x = , 2 t y = , 3 t z = , 1 t . Page 4 of 4 (c) r F d C where &gt; =&lt; xz y z 2 ), cos( , 2 F and C is any path in R 3 from the point (1,0,1) to the point 1 , 2 , 1 . (Use the Fundamental Theorem for Line Integrals.) (d) r F d C where &gt; + =&lt; 2 2 , xy e y x e y x F and C is the curve that runs along the upper half of the unit circle from (-1,0) to (1,0) followed by the line segment from (1,0) back to (-1,0). (Use Greens Theorem.) (e) dx z xy C + ) 7 ( where C is the line segment in R 3 from the point (1, 2, 3) to the point (4, 3, 2)....
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MAC2313_Test4_Spring2010 - F ). Page 3 of 4 6. (10 points...

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