T 1 dv test3mac2313 page 2 of 6 3 10 pts let f xyz xy

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T 1 dV
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TEST3/MAC2313 Page 2 of 6 _________________________________________________________________ 3. (10 pts.) Let F (x,y,z) = <xy 2 , yz 2 , zx 2 >. Compute the divergence and the curl of the vector field F . (a) div F = (b) curl F = _________________________________________________________________ 4. (10 pts.) Compute the surface area of the part of the graph of the surface defined by z = x + y 2 that lies above the triangle in the xy-plane given by the pairs (x,y) that satisfy 0 x 1 and x y 1. [Of course above means z 0. One order of integration may be easier than the other.]
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TEST3/MAC2313 Page 3 of 6 _________________________________________________________________ 5. (10 pts.) Write down a triple iterated integral in cartesian coordinates that would be used to evaluate , T f ( x , y , z ) dV where f(x,y,z) = x + y and T is the region contained between the two paraboloids z = 2 - x 2 - y 2 and z = x 2 + y 2 , but do not attempt to evaluate the triple iterated integral you have obtained. [Sketching the traces in the coordinate planes might help. Determining the projection of the intersection of the two surfaces on the xy-plane is essential.] _________________________________________________________________ 6. (10 pts.) Write down the triple iterated integral in
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