Riemann Sums and the Definite Integral

Approximating, then exact

Split $[a,b]$ into $n$ pieces of width $\Delta x=\frac{b-a}{n}$ and add up rectangles. Choosing heights at left endpoints gives the left sum, at right endpoints the right sum, at midpoints the midpoint sum. Using trapezoids instead gives

The definite integral is the limit of these sums as $n\to\infty$, and it equals the signed area: area below the axis counts as negative.

Over or under? For an increasing function, left sum underestimates and right sum overestimates. For a concave up function, the trapezoid rule overestimates and the midpoint rule underestimates. AP asks for the reason, so name the property (increasing / concave up), not just the verdict.