Line integral simplification from closed path integral

[Chandasouk]

http://img2.imageshack.us/img2/5061/14983795.jpg

I have no idea how they simplified the integral to the second step.

 

[LCKurtz]

That looks like a line integral along a parameterized curve c(t) = <x(t),y(t),z(t)>.

You didn't copy the part that gives the formula for c(t). And remember that

ds = ||c'(t)|| dt