Fundamental Theorem for Line Integrals

[andyk23]

Vector field F(bar)= <6x+2y,2x+5y>
fx(x,y)= 6x+2y fy(x,y)= 2x+5y
f(x,y)= 3x^2+2xy+g(y)
fy(x,y)=2x+g'(y)
2x+g'(y)= 2x+5y
g'(y)= 5y
g(y)= 5/2*y^2
f(x,y)=3x^2+2xy+(5/2)y^2
Then find the \int F(bar)*dr(bar) along curve C t^2i+t^3j, 0<t<1
I'm stuck on finding the last part for the F(bar) would I use <6x+2y,2x+5y> and substitute the <t^2,t^3> in the for x&y then do F(bar)* rbar'(t)
so \int from 0 to 1 of <6t^2+2t^3,2t^2+5t^3> <2t,3t^2>
Thanks for the help

 

[Dick]

You figured out a potential for your vector field. You could use that to do the integral. And then you set up the explicit contour integral. Both look correct. They give you the same answer, don't they?