AP Calculus AB and BC glossary

Overestimate and underestimate

An approximation of a definite integral overestimates when it exceeds the true value and underestimates when it falls short. Left and right Riemann sums are decided by whether the function is increasing or decreasing; the trapezoidal and midpoint rules are decided by concavity.

For a monotonic function the endpoint sums are clean to call. On an increasing function the left sum draws each rectangle's height from the lower left edge, so it underestimates, while the right sum overestimates. On a decreasing function the two roles swap.

f increasing on [a,b]:Ln≤∫abf(x) dx≤Rnf \text{ increasing on } [a,b]:\quad L_n \le \int_a^b f(x)\,dx \le R_n

Concavity, not monotonicity, governs the trapezoidal and midpoint rules. On a concave up curve the connecting chords lie above the graph, so the trapezoidal rule overestimates while the midpoint rule underestimates. A concave down curve reverses both.

The mistake

Treating "the left sum underestimates" as a law. It holds only while the function is increasing; a decreasing function makes the left sum overestimate instead. The left sum has a single, predictable direction only while the function is monotonic, so if ff rises then falls across the interval it has no fixed direction at all. And it is monotonicity that governs left and right sums; concavity is what governs trapezoids and midpoints, not the other way around.

Appears in: Unit 6: Integration and Accumulation