Integral Over CFds: Homework Statement & Eqns

[dylanhouse]

Homework Statement

For F = (x^2+y)i + (y-x)j, calculate the integral over C of Fds for r = (t, t^2) where t goes from 0 to 1.

Homework Equations

The Attempt at a Solution

I know the integral over C of fds is f*sqrt(r'(t)^2+r^2*theta'(t)^2) dt. But I have no theta in this question, is this the wrong integral?

 

[LCKurtz]

Homework Statement

For F = (x^2+y)i + (y-x)j, calculate the integral over C of Fds for r = (t, t^2) where t goes from 0 to 1.

Homework Equations

The Attempt at a Solution

I know the integral over C of fds is f*sqrt(r'(t)^2+r^2*theta'(t)^2) dt. But I have no theta in this question, is this the wrong integral?

Yes, that's the wrong integral. This problem has nothing to do with polar coordinates. You have $\vec r(t) =\langle x(t),y(t)\rangle = \langle t,t^2\rangle$. Write the integral in terms of $t$.

 

[dylanhouse]

So F would be (2t^2)i + (t^2 - t)j. And dr is just <1, 2t>dt. But how would I integrate with respect to ds if I end up with a dt?

 

[LCKurtz]

So F would be (2t^2)i + (t^2 - t)j. And dr is just <1, 2t>dt. But how would I integrate with respect to ds if I end up with a dt?

Your integral is stated with an Fds. I don't think that is an arc length integral, but then you haven't told us what that notation means. F is a vector. What is ds? I would assume you mean $\vec F\cdot d\vec s$ which might otherwise be written $\vec F\cdot \hat T~ds$ or $\vec F\cdot d\vec r$. In any case I expect you would evaluate it as $\int\vec F\cdot \vec r'(t)~dt$.