How did they get from (1) to (2) in this picture? It says "it is easily shown", but I don't quite see it
You don't go from (1) to (2). You go from the first of your integrals to (2).
Picture it as the area under a curve and slide this say p units to the right.
The area doesn't change, the limits now go from p-h to p+h and the approximation becomes
$\displaystyle h\Big[f\Big(p-\frac{h}{\sqrt{3}}\Big)+f\Big(p+\frac{h}{\sqrt{3}} \Big)\Big]$.
Now put p-h = a, p+h = b, and solve for p and h in terms of a and b.