# Point 2 1 2 b also find parametric equations for the

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point (2 , - 1 , 2). (b) Also find parametric equations for the normal line to the graph from (a) at (2 , - 1 , 2).

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EGR 265, Spring 2012, Final Exam 8 Problem 7 (8 points) Find the line integral Z C xy 2 ds, where C is a quarter of a unit circle centered at the origin and contained in the first quadrant, starting at (1, 0) and ending at (0, 1).
EGR 265, Spring 2012, Final Exam 9 Problem 8 (11 points) (a) Verify that the force field F ( x, y ) = (2 xy - y 2 + 1) i + ( x 2 - 2 xy ) j is conservative. (b) Find a potential function φ ( x, y ) for F ( x, y ). (c) Find the work done by the force field F ( x, y ) along the curve x = 2 /t , y = t 2 , 1 t 2.

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EGR 265, Spring 2012, Final Exam 10 Problem 9 (11 points) A lamina of constant density ρ ( x, y ) = 1 is bounded by the curves y = x 2 and y = 1. (a) Find the lamina’s mass. (b) Find the lamina’s centroid. Use geometric considerations to simplify your work.
EGR 265, Spring 2012, Final Exam 11 Problem 10 (8 points) Rewrite the function f ( x, y ) = x + y using polar coordinates and find its integral over the quarter disk of radius 1 in the first quadrant.

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Scratch Paper 12
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• Fall '12
• Franklin
• Constant of integration, Boundary value problem

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