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Evaluate the line integral \[\int_c F\bullet dr\]
where C is given by the vector function r(t).
\[F(x,y,z)=(x+y)\hat{i}+(yz)\hat{j}+z^2\hat{k}\]
\[r(t)=t^2\hat{i}+t^3\hat{j}+t^2\hat{k}\]
\[0 \le t \le 1\]
\[r'(t)=2t\hat{i}+3t^2\hat{j}+2t\hat{k}\]
\[\int_c F \bullet dr=\int_a^b F(r(t)) \bullet r'(t)dt\]
\[\int_0^1 (t^2\hat{i}+(t^3t^2)\hat{j} + t^4\hat{k})\bullet (2t\hat{i} +3t^2\hat{j}+2t\hat{k})dt\]
how am I doing?
 one year ago
 one year ago
Evaluate the line integral \[\int_c F\bullet dr\] where C is given by the vector function r(t). \[F(x,y,z)=(x+y)\hat{i}+(yz)\hat{j}+z^2\hat{k}\] \[r(t)=t^2\hat{i}+t^3\hat{j}+t^2\hat{k}\] \[0 \le t \le 1\] \[r'(t)=2t\hat{i}+3t^2\hat{j}+2t\hat{k}\] \[\int_c F \bullet dr=\int_a^b F(r(t)) \bullet r'(t)dt\] \[\int_0^1 (t^2\hat{i}+(t^3t^2)\hat{j} + t^4\hat{k})\bullet (2t\hat{i} +3t^2\hat{j}+2t\hat{k})dt\] how am I doing?
 one year ago
 one year ago

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MathSofiyaBest ResponseYou've already chosen the best response.3
Oops wrong question, it's \[F(x,y,z)=z\hat{i}+y\hat{j}z\hat{k}\] \[r(t)=t\hat{i}+sint\hat{j}+cost\hat{k}\] \[0 \le t \le pi\]
 one year ago

Outkast3r09Best ResponseYou've already chosen the best response.0
alright guess we'll start over hahha continues
 one year ago

MathSofiyaBest ResponseYou've already chosen the best response.3
would that make \[r'(t)=\hat{i}+cost\hat{j}sint\hat{k}\] and
 one year ago

MathSofiyaBest ResponseYou've already chosen the best response.3
\[\int_0^{\pi} (t\hat{i}+sint\hat{j}+cost\hat{k})\bullet (\hat{i}+cost\hat{j}sint\hat{k})dt\]
 one year ago

KingGeorgeBest ResponseYou've already chosen the best response.1
So far that looks right to me.
 one year ago

KingGeorgeBest ResponseYou've already chosen the best response.1
You're welcome. Once you do the dot product, it looks like it will be very easy to evaluate.
 one year ago

KingGeorgeBest ResponseYou've already chosen the best response.1
So can you show me what you think the dot product will be?
 one year ago

MathSofiyaBest ResponseYou've already chosen the best response.3
I can't just multiply the i j and k coefficients can I? \[t+sintcostsintcost\]
 one year ago

KingGeorgeBest ResponseYou've already chosen the best response.1
That's exactly what it is. You should end up with \[t+\sin(t)\cos(t)\sin(t)\cos(t)=t\]
 one year ago

MathSofiyaBest ResponseYou've already chosen the best response.3
Yaaaaayyyyy!!!!! I just need to build confidence i guess.... sigh
 one year ago

KingGeorgeBest ResponseYou've already chosen the best response.1
You'll get used to it.
 one year ago

MathSofiyaBest ResponseYou've already chosen the best response.3
here is the mistake I see: x=t y=sint z=cost that would give us \[\int_0^{\pi} cost dt+\int_0^{\pi}sintcostdt+\int_0^{\pi}tsint dt\]
 one year ago

TuringTestBest ResponseYou've already chosen the best response.0
\[\vec F(x,y,z)=z\hat{i}+y\hat{j}z\hat{k}\]\[\vec r(t)=t\hat i+\sin t\hat j+\cos t\hat k\]\[\vec r'(t)=\hat i+\cos t\hat j\sin t\hat k\]\[\int_0^\pi\vec F(\vec r(t))\cdot\vec r'(t)dt=\int_0^\pi\langle\cos t,\sin t,\cos t\rangle\cdot\langle1,\cos t,\sin t\rangle dt\]do you see a problem with that?
 one year ago

TuringTestBest ResponseYou've already chosen the best response.0
\[=\int_0^\pi\cos t+\sin t\cos t\sin t\cos tdt=\int_0^\pi\cos tdt=2\]
 one year ago

MathSofiyaBest ResponseYou've already chosen the best response.3
oh I see what I did! It's z y z and not z y x
 one year ago

MathSofiyaBest ResponseYou've already chosen the best response.3
Have a good night's rest!
 one year ago
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