notes001 (26)

# notes001 (26) - d r = Z C r f ± d r = f(1 1& f(0 0...

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November 20, 2009 NAME________________________________ Let F be the vector ±eld F ( x; y ) = 32 j and let C be the parabolic curve y = x 2 beginning at the point (0 ; 0) and ending at the point (1 ; 1) . Evaluate the line integral Z C F ± d r . You must include all details of your work. Solution: The easiest way to do this is to use the Fundamental Theorem of Line Integrals: The vector ±eld F ( x; y ) = 32 j is conservative and has potential function f ( x; y ) = 32 y . Thus Z C F ±
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Unformatted text preview: d r = Z C r f ± d r = f (1 ; 1) & f (0 ; 0) = & 32 & 0 = & 32 . Now we show how to do the integral using the de±nition of line integrals: The curve C can be parameterized as r ( t ) = t i + t 2 j ² t ² 1 from which we see that r ( t ) = i + 2 t j and F ( r ( t )) ± r ( t ) = ( & 32 j ) ± ( i + 2 t j ) = & 64 t . Thus Z C F ± d r = Z C F ( r ( t )) ± r ( t ) dt = Z 1 & 64 t dt = & 32 . 1...
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