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randolph4

# randolph4 - F x y z = h xz x 2 y i(a Find curl F(b Find div...

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Math 22 Exam #4 Name Fall 2000 1. [9pt] Evaluate the line integral: C ( x + y 2 ) ds where C is the straight line from (0 , 0) to (2 , 4) then from (2 , 4) to (0 , 4). 2. [12pt] Find the work done by the vector field F ( x, y ) = xy, y as a particle moves along the parabola y = x 2 from (0 , 0) and (2 , 4). 3. [10pt] Find the value of the line integral C x 4 dx - xy dy where C is the closed curve beginning at (0,0) and running counterclockwise around the triangle whose vertices are (0 , 0), (2 , 4), (0 , 4). 4. [13pt] Let f ( x, y ) = 2 y 2 + x 2 y . (a) Calculate F = f . (b) For F as above, find the value of C F · d r where C is the curve beginning and ending at (0,0) along the path r ( t ) = 1 - cos( t 2 ) , sin( t 2 ) , 0 t 2 π (think before you write) (c) For F as above, find the value of C F · d r where C is the curve from (0,0) to (0,1) along the path r ( t ) = 1 - cos( t 2 ) , sin( t 2 ) , 0 t π/ 2 5. [20pt] Consider the vector field
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Unformatted text preview: F ( x, y, z ) = h xz, , x 2 y i (a) Find curl F (b) Find div F (c) Find โ ยท ( โ ร F ) (d) Is there a potential function, f , for which โ f = F ? Explain. 6. [10pt] Find the area of the surface which is described by the parametric equations x = uv, y = u + v, z = u-v , where u 2 + v 2 โค 1. 7. [10pt] SET UP (donโt evaluate) an iterated integral (include the appropriate limits of integration) whose value gives the area of the surface z = x 2 + 3 y that lies inside the cylinder x 2 + y 2 = 4. 8. [16pt] Evaluate the following surface integral : ZZ S y dS where S is the the part of the cylinder y 2 + z 2 = 4 that lies between x = 1 and x = 4....
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