samplefinal_001 - MAC 2313 Final Exam(counts for 40 of...

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MAC 2313 Final Exam (counts for 40% of course grade) Time: Two hours. The best five solutions will count. Please show all working. Only a fraction of the total points on each problem are for the final answer itself. Read through the problems before you begin to write. They are not in order of topics or difficulty. At the bottom of the exam there is attached a list of theorems/formulae, without explanations of the symbols, which may be helpful for certain questions. 1. Find the center of mass of the lamina bounded by the graph of y = 9 - x 2 and y = 0, if the density is given by ρ ( x, y ) = ky 2 . 2. Use Green’s theorem to evaluate the work done in moving around the circle x 2 + y 2 = 4 through the force field given by F ( x, y ) = ( xy, x + y ) . 3. Use the divergence theorem to calculate the flux integral of the vector field F ( x, y, z ) = ( xe z + y 7 , ( y + x 5 ) e z , e z ) through the surface of the solid bounded by z = 4 - y , z = 0, x = 0, x = 6 and y = 0. 4. Sketch the curve C with parametric equations x = a cos t, y = b sin t, 0 t 2 π.
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