Use the defiinition of a line integral to evaluate

[Bamboozled91]

∫

Homework Statement

Use the definition to find the line integral of F(x,y) = (y,x) along each of the following paths.

The parabola y = x^2 from (-1,1) to (1,1)

Homework Equations

F(x) = gradientf(x)

∫F(x) dx = f(b) - f(a)

The Attempt at a Solution

I tried (y,x) dot (t,t^2) which gave me yt+xt^2 which 2t^3 thus ∫ from 1 to -1 of 2t^3 unfortunatley this was incorrect so I just did this r(t) = (1-t)<-1,1> + t<1,1> which give
<-1,1+2t> then ∫x^2 ds = ∫(-1)^2sqrt(4) = ∫2 dt = 2t from 0 to 1 which gives 2 this was correct however I am not sure if this is a valid way to answer

 

[LCKurtz]

∫

Homework Statement

Use the definition to find the line integral of F(x,y) = (y,x) along each of the following paths.

The parabola y = x^2 from (-1,1) to (1,1)

Homework Equations

F(x) = gradientf(x)

∫F(x) dx = f(b) - f(a)

The Attempt at a Solution

I tried (y,x) dot (t,t^2) which gave me yt+xt^2 which 2t^3 thus ∫ from 1 to -1 of 2t^3 unfortunatley this was incorrect so I just did this r(t) = (1-t)<-1,1> + t<1,1> which give
<-1,1+2t> then ∫x^2 ds = ∫(-1)^2sqrt(4) = ∫2 dt = 2t from 0 to 1 which gives 2 this was correct however I am not sure if this is a valid way to answer

It isn't because that second method is along a straight line, not the parabola. But check that part in the red. Aren't you supposed to dot it with the derivative of < t, t²>?

 

[Bamboozled91]

I will try that sorry about the late reply

 

[Bamboozled91]

Lol your right I goofed it thanks man.