Evaluate the line integral, where C is the given curve. integral C

uneskovogl5 2021-11-16 Answered
Evaluate the line integral, where C is the given curve. integral C xeyzds C is the line segment from (0,0,0) to (1,2,3)
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Expert Answer

Jennifer Hill
Answered 2021-11-17 Author has 10 answers

Step 1
Let C the line segment from (0,0,0) to (1,2,3). The parametric equation for this line in vector form is
r(t)=r0+td
where we chose r0=0,0,0r and the direction vector d is
d=1,2,30,0,0=1,2,3
Hence we have
r(t)=t,2t,3t
and we have in scalar form the parametric equations
for C :
x=t, y=2t, z=3t, 0t1
To detemine the line integral:
I=Cxeyzds
we first determine ds:
ds=(dxdt)2+(dxdt)2+(dzdt)2dt
=(1)2+(2)2=(3)2dt
=1+4+9dt
=14dt
Step 2
Cxeyzds=01te6t214dt=1401e6t2tdt
Now changing to the variable u=6t2 , we have du=12tdt=12tdt and:
I=1406112eudu=141206eudu
=1412eu60=1412(e6e0)
=1412(e61)
Finally:
cxeyz

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