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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Jennifer Hill · Accepted Answer

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=⟨0,0,0⟩r and the direction vector d is d=⟨1,2,3⟩−⟨0,0,0⟩=⟨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, 0≤t≤1 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=14∫01e6t2tdt Now changing to the variable u=6t2 , we have du=12tdt=12tdt and: I=14∫06112eudu=1412∫06eudu =1412eu60=1412(e6−e0) =1412(e6−1) Finally: ∫cxeyz

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

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