MAC
c3-fx(1)

# Use greens theorem to compute the work done on the

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. Use Green’s Theorem to compute the work done on the particle by the force field F ( x , y ) 8 xy ,4 x 2 10 x . ______________________________________________________________________ 4. (15 pts.) Find an arc-length parameterization for the curve that has the same orientation and has the point on the graph where t = 0 as the reference point. Rather than overloading the symbol r , write this new parameterization as R (s). If you move along the curve using the parameterization given by R , what’s your speed?

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Name: Final Exam/MAC2313 Page 3 of 8 ______________________________________________________________________ 5. (15 pts.) Evaluate the iterated integral by first converting the integral to an equivalent integral in polar coordinates, and then evaluating the iterated integral you obtain. A picture of the region helps. ______________________________________________________________________ 6. (15 pts.) Write down but do not attempt to evaluate iterated triple integrals in (a) rectangular, (b) cylindrical, and (c) spherical coordinates that would be used to compute the volume of the sphere defined by x 2 + y 2 + z 2 = (5) 2 . [For rectangular, there are many correct variants.] (a) V = (b) V = (c) V =
Name: Final Exam/MAC2313 Page 4 of 8 ______________________________________________________________________ 7. (15 pts.) Determine the absolute maximum and minimum values of f ( x , y ) = x 2 + 2 y 2 - x and where they are obtained when ( x , y ) lies in the closed disk defined by x 2 + y 2 4. Analyze f on the boundary by using Lagrange Multipliers, but don’t forget the interior.

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