AP Calculus AB and BC glossary

Riemann sum

Also called: Left Riemann sum, Right Riemann sum, Midpoint sum

A Riemann sum approximates a definite integral by slicing the interval into subintervals and adding up the area of a rectangle on each one. As the number of subintervals grows without bound, the sum approaches the exact value of the integral.

\sum_{i=1}^{n} f(x_i^*)\,\Delta x, \qquad \Delta x = \frac{b-a}{n}

The only choice that distinguishes the standard versions is where each rectangle takes its height: the left endpoint, the right endpoint, or the midpoint of its subinterval. The trapezoidal rule uses trapezoids instead and equals the average of the left and right sums.

For an increasing function the left sum underestimates and the right sum overestimates, and for a decreasing function the roles swap. Concavity decides the trapezoidal and midpoint errors: trapezoids overestimate on a concave up curve.

The mistake

Using $n$ data points as $n$ subintervals. A table with values at 0, 2, 5, and 9 gives three subintervals of unequal width, so $\Delta x$ is not constant.

Appears in: Unit 6: Integration and Accumulation