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# Y in the plane b using the fundamental theorem of

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y ) in the plane. (b) Using the Fundamental Theorem of Line Integrals, evaluate the path integral below, where C is any smooth path from the origin to the point ( π /2,ln(2)). [ WARNING: You must use the theorem to get credit here.] ______________________________________________________________________ 8. (18 pts.) Write down but do not attempt to evaluate the iterated triple integrals in (a) rectangular, (b) cylindrical, and (c) spherical coordinates that would be used to compute the volume of the sphere with a radius of 1 centered at the origin. [For rectangular, there are many correct variants.] (a) V = (b) V = (c) V =

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Name: Final Exam/MAC2313 Page 5 of 7 ______________________________________________________________________ 9. (12 pts.) Let . Compute the divergence and the curl of the vector field F . (a) div F = (b) curl F = ______________________________________________________________________ 10. (16 pts.) Use the substitution u = x + y , v = x - y to evaluate the integral where R is the region enclosed by the lines x + y = 0, x + y = 1, x - y = 1, and x - y = 4
Name: Final Exam/MAC2313 Page 6 of 7 ______________________________________________________________________ 11. (12 pts.) A particle, starting at (0,0) moves to the point (1,1) along the parabola y = x 2 and then returns to (0,0) along the parabola x = y

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y in the plane b Using the Fundamental Theorem of Line...

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