AP Calculus AB and BC

Riemann Sum vs Definite Integral

Q: How many subintervals should I use?

The problem tells you. AP questions specify both n and the sample point, because the value of the estimate depends on both. Only the limit is independent of those choices.

A Riemann sum is a finite approximation: it adds the areas of n rectangles across n subintervals. The definite integral is the exact value those sums approach as n goes to infinity and the subinterval width goes to zero. One is an estimate you can compute by hand, the other is the limit it converges to.

Riemann sum

Use when: The problem gives a table of values, fixes the number of subintervals, or uses the word approximate, so you add up finitely many rectangles.

Definite integral

Use when: You want the exact accumulated amount and have an integrand you can antidifferentiate, or a calculator-active problem where the numerical value is what is scored.

Side by side

	Riemann sum	Definite integral
What it measures	An approximation of the accumulation on $[a,b]$	The exact accumulation on $[a,b]$
Written as	$\sum_{i=1}^{n} f(c_i)\,\Delta x$	$\int_a^b f(x)\,dx$
Number of pieces	A finite $n$ the problem chooses	The limit as $n \to \infty$
When to reach for it	Tabulated data, or a required left, right, or midpoint estimate, or a trapezoidal sum built from them	A closed form integrand you can antidifferentiate, or a calculator-active question wanting the value to three decimals
Common trap	Using $\Delta x = b - a$ instead of $\frac{b-a}{n}$	Reading it as area when the integrand dips below the axis

The two are not rivals. The definite integral is defined as the limit of Riemann sums, so the sum is the construction and the integral is what the construction converges to. For a continuous $f$ on $[a,b]$ every choice of sample point gives the same limit, which is why the definition does not care whether you take left endpoints, right endpoints, or midpoints.

\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(c_i)\,\Delta x, \qquad \Delta x = \frac{b-a}{n}

Reading the direction of the error is the part the exam tests. On an interval where $f$ is increasing, a left sum underestimates and a right sum overestimates; on a decreasing interval those swap. Concavity governs the other pair: the trapezoidal sum overestimates where $f$ is concave up, and the midpoint sum underestimates there.

The mistake

Writing an approximation with an equals sign. Four rectangles do not produce $\int_0^8 f(x)\,dx$ , they produce an estimate of it. Say approximately, and name the sum you used, because the reader cannot check a number whose method is hidden.

Frequently asked questions

Is a Riemann sum ever exact?

Yes, when the geometry cooperates. Any sum is exact for a constant $f$ , and the midpoint sum is exact whenever $f$ is linear, as is the trapezoidal sum, which is not itself a Riemann sum but the average of the left and right ones. Otherwise a finite sum carries error that shrinks as $n$ grows.

How many subintervals should I use?

The problem tells you. AP questions specify both $n$ and the sample point, because the value of the estimate depends on both. Only the limit is independent of those choices.

In the CED: Unit 6: Integration and Accumulation