geometric meaning behind line integrals

Question

What are some geometric meanings behind line integrals? I know if you have a curve on the xy plane and you are given a function $f(x,y)$ then the geometric meaning is a "curtain drawn" from the function (surface) to the curve below. This is a planer curve.

However what about when we just have a helix? Similarly what the geometric meaning (or explanation) for the line integral involving a vector field $F$ and a curve. $$\int_c{F\cdot ds}$$

Here is an example:

Let $c(t)= \sin{t}, \cos{t}, t$ from 0 to $2\pi$. let the vector field $F$ be defined by $$F(x,y,z)=x\hat{i} + y\hat{j} + z\hat{k}$$ Compute $\int_c{F\cdot ds}$

Any links to pdfs and other resources helping me understanding would be very helpful!

Thanks!

Answer 1

There are several really good resources. But for all of your calculus needs, Paul's Online Notes is typically very well written. The link for his beginning of an explanation of line integrals is http://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtI.aspx.

Also, Paul does a very good job at explaining many other facets of calculus. Enjoy.