The work integral (general question)

[ZanyCat]

I'm sure you're all familar with this formula for work, $W=\int_{s_i}^{s_f} \mathrm{\vec{F}}\cdot\,\mathrm{d}\vec{s}<br />$

I don't understand how to evaluate this integral. How do you antidifferentiate in terms of a vector? How do you evaluate the dot product when ds isn't an actual value?

Thanks! :)

 

[NaOH]

It's done with something called vector calculus, more specifically, line integrals.
http://en.wikipedia.org/wiki/Line_integral#Vector_calculus

Depending on whether it's a scalar or vector field, it is calculated slightly differently. Nevertheless, we will do a dot product inside the integral while manipulating the integral into something more familiar, so only parallel components will count.

 

[ZanyCat]

So would we do the dot product of F and delta-s? If so, that would give us a scalar value (or perhaps a function), and what would we integrate that in terms of?

 

[SteamKing]

Remember that ds can be broken down into components dx and dy just like F can be broken down into components Fx and Fy (Fx and Fy do not represent partial derivatives here).