Curvature and Tangential angle

[ForMyThunder]

In Differential Geometry by Heinrich Guggenheimer (if you have the book, the proof I am asking about is Theorem 2-19), he gives the angle between a chord through points s and s' and the tangent at s, as the integral of the curvature (with respect to arc length) from s' to s. I'm not sure how he got this, because I thought that the integral of curvature gave the angle between the tangent and the x-axis. Or maybe I'm not understanding something.

 

[Eynstone]

It should be the angle between the tangents at the endpoints. (You can, of course, calculate the slope of the chord thereon).

 

[ForMyThunder]

Here's the book from Google books. The proof is on page 31.

http://books.google.com/books?id=CS...&resnum=4&ved=0CCkQ6AEwAw#v=onepage&q&f=false