Students will tell you that these aren't great methods independently but pretty good taken together. After all, you're adding the left and right endpoints and dividing by 2. That's how you find the area of a trapezoid. Another alternative that students suggest is to use the midpoint of the interval. Also, errors are going down by a factor of 24 as n increases by a factor of 5.
Midpoint rule is equivalent to a trapezoid whose top is tangent to the curve at the midpoint.
There's something better. This gives Simpson's Rule. Well, you do have to be smart about how you pick your rules. Just because the technology is out there to do the number crunching doesn't release the human from devising efficient methods. I don't know of a simple geometric interpretation for this.