Left, Right and Midpoint Sums

To build each rectangle, we must choose where to read the height $f(x_k)$ on the piece. Three classic choices:

Sum	Height of the rectangle
left	$f$ at the left endpoint of each piece
right	$f$ at the right endpoint of each piece
midpoint	$f$ at the middle of each piece

For an increasing function, these choices bracket the actual area: the left rectangles are too low and underestimate the area, while the right ones are too high and overestimate it.

$$\sum_{k} f(x_{k-1})\,\Delta x \;\leq\; \int_a^b f(x)\,\mathrm{d}x \;\leq\; \sum_{k} f(x_k)\,\Delta x \qquad (f \text{ increasing})$$

The exact area is thus sandwiched between the two — an excellent way to obtain bounds.

The midpoint, for its part, balances the overshoots on either side and generally gives a much better approximation, even with few rectangles.

Whatever the choice, a reassuring result: as $n$ tends to infinity, all three sums converge to the same limit, the integral. The choice of point only affects the speed of approach, not the final value. This is what makes the definition of the integral solid and unambiguous.