A More Accurate Riemann Sum

The method of taking a Riemann sum using rectangles is really sloppy mathematically speaking. When using left-hand or right-hand limits there is unaccounted for area or overlap. The error on this method is dramatically larger than that of other polygonal methods.

Of course since it’s a limit at infinity, this really doesn’t matter, but I personally would like to see other methods.
Less established ones. Not Simpson’s rule or any of that.

I propose coming up with the theory of taking a Riemann Sum using iterations of pairs of right triangles.
It would effectively work the same way, but you can maximize the right triangles to take up more area on each iteration, so the compound error doesn’t build up as quick as the limit of the sum approaches infinity.

However, again, because Riemann Sums are limits at infinity, none of this really makes a difference. All the established methods are infinitely accurate.

One method for a more accurate Riemann Sum

Of course, in retrospect this method sounds a lot like the method of exhaustion that was utilized by the Ancient Greeks.