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What is the solution to this? I've never seen this notation before.

 

 Oct 20, 2022
 #1
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ANSW = 0 with a domain x all -1,1 and y all 0 to infinity

 Oct 20, 2022
 #2
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Same with me, I've not seen this notation before, so this could be wrong.

I think that the e's are unit vectors and that we are being given the parametric equations of the line of integration.

\(x=u^{2},\quad y=2u, \quad z=u^3\)

from which

\(2xyz^{2}=2.u^{2}.2u.(u^{3})^{2}=4u^{9},\)

and

\(dr=e_{x}dx+e_{y}dy+e_{z}dz=e_{x}2u\, du+e_{y}2\, du+e_{z}3u^{2} du\)

So,

\(\displaystyle \int_{L}2xyz^{2}\, dr=e_{x}\int^{1}_{0}4u^{9}.2u\,du+e_{y}\int^{1}_{0}4u^{9}.2\, du+e_{z}\int^{1}_{0}4u^{9}.3u^{2}\,du \\ =(8/11)e_{x}+(4/5)e_{y}+e_{z}.\)

Apologies if that's a load of rubbish !

 Oct 20, 2022

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