- #1
Telemachus
- 835
- 30
Homework Statement
Hi. I have a doubt with this exercise. I'm not sure about what it asks me to do, when it asks me for the curvilinear integral. The exercise says:
Calculate the next curvilinear integral:
[tex]\displaystyle\int_{C}^{}(x^2-2xy)dx+(y^2-2xy)dy[/tex], C the arc of parabola [tex]y=x^2[/tex] which connect the point [tex](-2,4)[/tex] y [tex](1,1)[/tex]
I've made a parametrization for C, that's easy: [tex]\begin{Bmatrix} x=t \\y=t^2\end{matrix}[/tex] [tex]\begin{Bmatrix} x'(t)=1 \\y'(t)=2t\end{matrix}[/tex]
And then I've made this integral:
[tex]\displaystyle\int_{-2}^{1}t^2-2t^3+(t^4-2t^3)2t dt[/tex]
But now I'm not too sure about this. What I did was:
[tex]\displaystyle\int_{a}^{b}F(\sigma(t))\sigma'(t)dt[/tex]
But now I don't know if I should use the module, I did this: [tex]\displaystyle\int_{a}^{b}F(\sigma(t))\sigma'(t)dt[/tex] and I don't know when I should use this: [tex]\displaystyle\int_{a}^{b}F(\sigma(t)) \cdot ||\sigma'(t)||dt[/tex]
I mean, both are curvilinear integrals, right?
I think that I understand what both cases means, but I don't know which one I should use when it asks me for the "curvilinear integral". The first case represents the area between the curve and the trajectory, and the second case represents the projection of a vector field over the trajectoriy, i.e. the work in a physical sense, but I know it have other interpretations and uses.
Well, that's all. Bye there, thanks for posting.