Well, it would help if you told us what , , and mean!
I think he's using Simpson's rule. If I'm correct, it looks like he's trying to show the relation between the midpoint rule and the trapezoidal rule with Simpson's rule.
Midpoint rule:
Trapezoidal rule:
Weighted average:
This weighted average is Simpson's rule.
Simpson's rule:
Looking at his equation,
= weighted average = Simpson's Rule
Not totally sure what these subscripts are, but there's a start.