Parameterize a triangle counter clockwise: F(x, y) = (−y + x2)i + (x − y2)j

[Colt87]

Find the line integral of a F of a triangle with vertices

F(x, y) = (−y + x2)i + (x − y2)j, and let R be the triangle with vertices (0, 0), (1, 0), and (0, 1).

I found the parameters in terms of t but don't know how to find r(t). I know that C = C1+C2+C3.

C1 = (t,0)
C2 = (1-t,t)
C3 = (0,1-t)

for 0<t<1

 

[Colt87]

I got the parameters as

C1 = (t,0)
C2 = (1-t,t)
C3 = (0,1-t)

from 0<t<1

but i don't know how to find r(t).

 

[tkhunny]

I got the parameters as

C1 = (t,0)
C2 = (1-t,t)
C3 = (0,1-t)

from 0<t<1

but i don't know how to find r(t).

One problem. You get only one "0 <= t <= 1". All three of your segments run simultaneously. That's no good. You must run them consecutively. In other words, if the first segment stops at t = 1, the second segment must start at t = 1. It doesn't get to start over at t = 0.

Give it another go.

 

[Colt87]

One problem. You get only one "0 <= t <= 1". All three of your segments run simultaneously. That's no good. You must run them consecutively. In other words, if the first segment stops at t = 1, the second segment must start at t = 1. It doesn't get to start over at t = 0.

Give it another go.

OK so then it would go

C1 - 0 <= t <=1
C2 - 1 <= t <= 1-x
C3 - 1-x <= t <= 0

would that be correct?