MathSofiya
Evaluate the line integral \[\int_c F\cdot dr\] where c is the vector function
\[\hat{r}(t)=t^3\hat i-t^2 \hat j+ t \hat k\]
and
\[F(x,y,z)=sinx\hat i + cosy \hat j+zx \hat k\]
Here is how far I have gotten so far:
\[\hat{r}'(t)=3t^2\hat i-2t \hat j+ \hat k\]
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lgbasallote
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you might want to use \cdot rather than \bullet hehe
MathSofiya
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I'll change the Function F in terms of t's in just a second
MathSofiya
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thanks @lgbasallote. Better? :P
lgbasallote
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yes. yes it is.
MathSofiya
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Hey i think I got it!
\[x=t^3;y=-t^2;z=t\]
YES?
MathSofiya
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That would give me:
\[F(r(t))=sin(t^3)\hat i +cos(-t^2)\hat j+t^4 \hat k \]
MathSofiya
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\[\int_{t=0}^{t=1}\left(sin(t^3)\hat i +cos(-t^2)\hat j+t^4 \hat k\right)\cdot\left(3t^2\hat i -2t\hat j +\hat k \right) dt\]
MathSofiya
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and now I would just do the dot product correct?
mukushla
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exactly
MathSofiya
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\[\int_0^1(3t^2sin(t^3)-2tcos(-t^2)+t^4)dt\]
mukushla
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well done
MathSofiya
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wait, I have to take the integral of several products?
mukushla
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separate them\[\int_{0}^{1} 3t^2 \sin t^3 dt+...+...\]
MathSofiya
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oh ok
\[\int_0^1 3t^2sin(t^3)dt-\int_0^12tcos(-t^2)dt+\int_0^1t^4dt\]
like this?
mukushla
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yep
MathSofiya
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oh integration by parts! duh =D
mukushla
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\[-\cos t^3\] :-)
TuringTest
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no integration by parts
you can do it all with u-subs
TuringTest
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do you get that?
MathSofiya
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Oh I see it now!!!!
TuringTest
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cool :)
MathSofiya
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u=t^2
du=2t dt for the second integral
TuringTest
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yep
MathSofiya
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u=t^3
and du=3t^2 dt for the first integral...LOL that took me a while!
TuringTest
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yeah, it's well set-up for the u-sub thing :)
mukushla
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u cooked the problem
MathSofiya
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\[-cos(1)+sin(1)+2\]
My algebra is probably wrong...but Yeah @mukushla that was one long recipe!
mukushla
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\[\large -\cos t^3]_{0}^{1}+\sin -t^2]_{0}^{1}+t^5/5]_{0}^{1}\]
MathSofiya
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\[(-cos(1)+1)+(-sin(1)+1)+\frac 1 5\]
\[-cos(1)-sin(1)+\frac{11}{5}\]
MathSofiya
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oops sine of 0 is zero
TuringTest
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you have an extra +1 in there - -yep
MathSofiya
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-cos(1)-sin(1)+ 6/5
TuringTest
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looks good to me :)
MathSofiya
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sigh...finally! Thanks guys!
TuringTest
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welcome !
mukushla
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:)