- #1
braindead101
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Compute the line integral [tex]\int_{C} F\cdot dr[/tex] where F = -y i + x j. The directed path C in the xy-plane consists of two parts: i) a left semicircle from (0, -1) to (0, 1) with center at the origin, and ii) a straight line segment from (0,1) to (2,1).
i) r(t) = cos t i + sin t j [pi/2 <=t<= 3pi/2]
ii) r(t) = 2t i + j [0<=t<=1]
for i):
F(r (t) ) = - sin t i + cos t j
r'(t) = - sin t i + cos t j
integ F(r(t)) dot r'(t) dt
= integ 1 dt [pi/2<=t<= 3pi/2]
so integral is
3pi/2 - pi/2 = pi ?
is this correct so far.
i) r(t) = cos t i + sin t j [pi/2 <=t<= 3pi/2]
ii) r(t) = 2t i + j [0<=t<=1]
for i):
F(r (t) ) = - sin t i + cos t j
r'(t) = - sin t i + cos t j
integ F(r(t)) dot r'(t) dt
= integ 1 dt [pi/2<=t<= 3pi/2]
so integral is
3pi/2 - pi/2 = pi ?
is this correct so far.