Line integral parameters
I'm looking at an example in my notes where the line integral of two segments is being evaluated so (integral)C=(integral)C1+(integral)C2.
For c1 : segment between (0,0) and (1,1)
For c2: part of the parabola y = sqrt(x) between (1,1) and (4,2)
for the parameterization it's written:
c1: x(t) = t, y(t) = t, 0<=t<=1
c2: x(t) = tē, y(t) = t, 1<=t<=2
Can someone explain to me how to get that parameterization?
For segment C1, just use the line segment to vector equation:
r(t) = (1-t)<Xo, Yo> + t<X1, Y1> where Xo,Yo are the starting points (0,0) and X1 and Y1 are the ending points (1,1).
For segment C2, note:
y = sqrt(x), so x = y^2. Thus, you can use y as the parameter.
y = t, x = t^2, and y goes from 1 to 2.
hope this helps