I'm sitting in on a course in dynamics and I've noticed something very strange that I can't get my mind around. So when they do problems that involve springs, they are saying that the work is $$U_s=-\frac{1}{2}k(s_f-s_0)^2$$
However, I know that Hooke's Law says that the force of a spring is given by $F=-ks$, and so to find work we just integrate with respect to $s$:
$$U_s^*=-k\int_{s_0}^{s_f} s\: ds =-\frac{k}{2}(s_f^2-s_0^2) $$
Clearly $U_s\neq U_s^*$, but they continually use $U_s$ as the formula for work. Could anyone clarify why? Bellow is a link to an example of a problem that does this.
http://faculty.washington.edu/reinhall/ME230/Soultions_nonassigned/Chap18_19.pdf#page=3