MathSofiya
Evaluate the line integral \[\int_c F\cdot dr\]
where c is given by
\[\hat r (t)=t\hat i +sint\hat j +cost \hat k\]
\[0 \le t le \pi\]
and:
\[\hat r(t)=\hat i +cost \hat j -sint \hat k\]
\[F(x, y, z)=z\hat i+y \hat j-x\hat k\]
x=t y=sint z=cost
\[\int_0^{\pi}(cost\hat i +sint\hat j-t\hat k)\cdot(\hat i+cos(t) \hat j -sint \hat k)dt\]
\[\int_0^{\pi}(cost+sintcost+tsint)dt\]
\[\int_0^{\pi} cost dt+\int_0^{\pi} sintcostdt+\int_0^{\pi}tsintdt\]
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MathSofiya
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\[sint]_0^{\pi}+\int_0^0 udu+[-tcost+sint]_0^{\pi}\]
mukushla
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mukushla is not typing a reply…whats wrong with site...lol
let me check it
MathSofiya
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LOL...i think mukushla should type a reply
mukushla
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its allright but what is that middle integral !!?
MathSofiya
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u substitution
u =sint
du=cost dt
sin(pi)=0
sin(0)=0
mukushla
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almost done
MathSofiya
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0?
mukushla
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yes
MathSofiya
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cos Pi is -1 though
MathSofiya
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so negative pi should be the answer?
MathSofiya
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and cos (0) is 1 oh it cancels out!
MathSofiya
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no that still doesn't seem right... :(
mukushla
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i meant 0 for middle integral
mukushla
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answer is \(\pi\) ?
MathSofiya
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the first integral would be zero too
and then we have left
\[[-tcost+sint]_0^{\pi}=\pi\]
MathSofiya
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YAY!
mukushla
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We are DONE.....;-D