8 7 points a state the corollary to greens theorem

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8. [7 points] (a) State the corollary to Green’s Theorem which gives the area of a region as a line integral. (b) Use a line integral to find the area enclosed by the closed curve x 2 - 2 xy +3 y 2 = 1. 9. [10 points] For each of the following differential forms ω determine if ω is exact. If ω is exact, use the algorithm given in class to find its potential function g . (a) ω = (2 xyz + y 2 z ) dx + ( x 2 z + 2 xyz ) dy + ( x 2 y + xy ) dz (b) ω = (2 x sin z + ye x ) dx + ( e x + z ) dy + ( x 2 cos z + y + 2 z ) dz . 10. [15 points] Let S be the piece of the cylinder x 2 + y 2 = 1 which is above the xy –plane and below the plane z = 1 + x . (a) Find the equation of the tangent plane to S at the point parenleftbigg 1 2 , 1 2 , 1 parenrightbigg . (b) Find the surface area of S . (c) Evaluate integraldisplay S f dS , where f ( x, y, z ) = z . -1 0 1 x -1 0 1 y 0 1 2 z
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