2 y f1 dx f2 dy y sin x dx cos x dy 1 1 1 2 2

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Unformatted text preview: s not possible! z y x 3. We first note that the boundary γ = ∂R is only piecewise smooth, so we will parametrize it in three pieces γ = γ 1 + γ 2 + γ 3 and then sum the line integrals. To keep the region to the left we must parametrize the boundary in the counterclockwise direction (as shown in the sketch). π First piece: γ 1 (t) = (t, 0), 0 ≤ t ≤ . 2 y F1 dx + F2 dy = (y − sin x) dx + cos x dy = Π ,1 γ1 γ1 2 π /2 π /2 (0 − sin t)(1) + (cos t)(0) dt = − 0 sin t dt = Γ3 0 π /2 cos t 0 R Γ2 = −1. Second piece: γ 2 (t) = x π , t , 0 ≤ t ≤ 1. 2 Γ1 1 F1 dx + F2 dy = (y − sin x) dx + cos x dy = γ2 π ,1 Third piece: γ 3 (t) = 2 γ2 (t − 1)(0) + (0)(1) dt = 0. π , 1 , 0 ≤ t ≤ 1. 2 1 π π − + F1 dx+F2 dy = (y ...
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This document was uploaded on 03/17/2014 for the course MAT B42 at University of Toronto- Toronto.

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