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notes001 (80) - F with respect to C All details must be...

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MATH 2203 - Quiz 5 (Version 1) Solution April 15, 2011 NAME______________________ 1. For the curve C : y = 1 2 x 2 ; 0 x 2 , and the function f ( x; y ) = x 3 y , evaluate Z C f ( x; y ) ds . You must include all details (no missing steps). Solution: We can parameterize C by x = t; y = t 2 = 2 ; 0 t 2 . Thus r ( t ) = t i + 1 2 t 2 j and v ( t ) = r 0 ( t ) = i + t j and j v ( t ) j = p 1 + t 2 . This gives us Z C f ( x; y ) ds = Z 2 0 t 3 1 2 t 2 p 1 + t 2 dt = Z 2 0 2 t p 1 + t 2 dt = 10 p 5 ± 2 3 . 1
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F ( x; y ) = y i + x j and the curve C : r ( t ) = 2 cos ( t ) i + 2 sin ( t ) j 0 ± t ± 2 ,
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Unformatted text preview: F with respect to C . All details must be included. Solution: The ±ow is Z C F ² T ds = Z 2 & F ² r ( t ) dt = Z 2 & ( & 2 sin ( t ) i & 2 cos ( t ) j ) ² (2 sin ( t ) i + 2 cos ( t ) j ) dt = Z 2 & & & 4 sin 2 ( t ) & 4 cos 2 ( t ) ± dt = & 4 Z 2 & dt = & 8 & . 2...
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notes001 (80) - F with respect to C All details must be...

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