Question

evaluate line integral of (x+y) ds where C is the straight-line segment x=t, y=(1-t), z=0, from (0,1,0) to (1,0,0)

Answer 1

From (0, 1,0) to (1, 0,0) . That is when t =0 , x = 0, y =1, z =0; when t =1 , x =1, y =0, z=0
Hence, \(0\leq t\leq 1\)
Now the integral: (x +y) w.r.t t is x+y = t +1-t =1
Jacobian: \(\sqrt{(dx/dt)^2+(dy/dt)^2+(dz/dt)^2}dt \)

\(= \sqrt{(1)^2+(1)^2+(0)^2}dt =\sqrt{2}dt\)

Combine all
\[\int_0^1 \sqrt{2}dt\]
I think you can handle it from here.

Answer 2

thanks