Line Integral

Nov 2009
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Evaluate \(\displaystyle \int_Csinxdx+cosydy\) where C is the top half of the circle \(\displaystyle x^2+y^2=4\) from (2,0) to (-2,0) and the line segment from (-2,0) to (-3,2).

Haven't done vector calc in awhile, still a little bit rusty, can someone assist me with getting the parametric equations for the line segment?
 

Bruno J.

MHF Hall of Honor
Jun 2009
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Canada
The line segment from \(\displaystyle p_1\) to \(\displaystyle p_2\) can be parametrized as \(\displaystyle r(t)=(1-t)p_1+tp_2\), \(\displaystyle 0 \leq t \leq 1\).
 
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Jul 2007
894
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New Orleans
Evaluate \(\displaystyle \int_Csinxdx+cosydy\) where C is the top half of the circle \(\displaystyle x^2+y^2=4\) from (2,0) to (-2,0) and the line segment from (-2,0) to (-3,2).

Haven't done vector calc in awhile, still a little bit rusty, can someone assist me with getting the parametric equations for the line segment?

For the circle let

\(\displaystyle x=2\cos{t}\) and \(\displaystyle dx = -2\sin{t}\)

\(\displaystyle y =2\sin{t}\) and \(\displaystyle dy = 2\cos{t}\)

Now sub these values for

\(\displaystyle \int_Csinxdx+cosydy\)

and you get

\(\displaystyle C_1 = \int^{\pi}_{0} \bigg(-2\sin{t}\sin{(2\cos{t})}+ 2\cos{t}\cos{(2\sin{t})}\bigg)dt\)

For \(\displaystyle C_2\)

\(\displaystyle x = -2 -t\) and \(\displaystyle dx= -1\)

\(\displaystyle y =2t\) and \(\displaystyle dy = 2\)

and then just sub in like before.
 
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