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quiz8-solns

# quiz8-solns - Math 53 Solutions to Quiz 8 GSI Santiago Ca...

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Math 53 - Solutions to Quiz 8 GSI: Santiago Ca˜nez 1. Compute the line integral of the vector field F ( x, y, z ) = xy i + y 2 j + yz k over the line segment from (1 , 0 , - 1) to (3 , 4 , 2). Solution. Parametric equations for the line segment C in question are given by r ( t ) = h 1 + 2 t, 4 t, - 1 + 3 t i , 0 t 1 . Then tangent vectors are given by r 0 ( t ) = h 2 , 4 , 3 i , and so we have Z C F · d r = Z 1 0 F ( r ( t )) · r 0 ( t ) dt = Z 1 0 4 t (1 + 2 t ) , 16 t 2 , 4 t ( - 1 + 3 t ) · h 2 , 4 , 3 i dt = Z 1 0 [8 t (1 + 2 t ) + 64 t 2 + 12 t ( - 1 + 3 t )] dt = Z 1 0 (116 t 2 - 4 t ) dt = 116 3 - 2 = 110 3 . 2. Compute the line integral R C (4 x 3 y 2 - 2 xy 3 ) dx +(2 x 4 y - 3 x 2 y 2 +4 y 3 ) dy over the curve C below.

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Solution. First recall that the line integral in question is nothing but the line integral of the vector field F ( x, y ) = 4 x 3 y 2 - 2 xy 3 , 2 x 4 y - 3 x 2 y 2 + 4 y 3 over C . We claim this vector field is actually conservative, so we look for a function f ( x, y ) such that f x = 4 x 3 y 2 - 2 xy 3 and f y = 2 x 4 y - 3 x 2 y 2 + 4 y 3 . Integrating the first equation with respect to x gives f ( x, y ) = x 4 y 2 - x 2 y 3 + V ( y ) .
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quiz8-solns - Math 53 Solutions to Quiz 8 GSI Santiago Ca...

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