Riemann Sums, Summation Notation, and Definite Integral Notation
Translate partitions and sampled inputs between Riemann sums and definite integral notation.
College Board Topic 6.3 · CED p. 115 (PDF p. 120) · CED effective Fall 2020; project course outline 2026–27.
A Riemann sum is Σ f(xᵢ*)Δxᵢ: height times width, added over a partition. The sample xᵢ* lies in its own subinterval. For a continuous function, the limit as the largest width approaches zero is the definite integral. Uniform partitions use Δx=(b−a)/n; right endpoints are a+iΔx, and left endpoints are a+(i−1)Δx.
To translate a limit, find the width first, then the interval and the sampled input. For limₙ→∞ Σᵢ₌₁ⁿ (2/n)(1+2i/n)², the width is 2/n, the interval is [1,3], and the height is x². The limit is ∫₁³ x² dx. Going the other way, ∫₂⁵ cos x dx becomes limₙ→∞ Σᵢ₌₁ⁿ cos(2+3i/n)(3/n). For a general nonuniform partition, it is the maximum width approaching zero that matters.
Let's work through it
- Read Δx=2/n and xᵢ=1+2i/n. Reason: A right endpoint is a+iΔx.
- Write ∫₁³ x² dx = [x³/3]₁³ = 26/3. Reason: The interval has length 2, and the integral is the continuous Riemann-sum limit.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
What is Δx for n equal pieces of [−1,4]?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
What is Δx for n equal pieces of [−1,4]?
Worked answer and reason
5/n.
Width is interval length divided by number of pieces.
Retrieval 2
Does Σ f(xᵢ*) alone give a Riemann sum for area?
Worked answer and reason
No; include each width Δxᵢ.
The sum needs height times width.
Catch the tempting mistake
- n approaching infinity is enough for every partition.
- For arbitrary partitions, require the maximum width to approach zero. A partition could keep one large gap even while adding many tiny pieces elsewhere.
Calculator workspace and limits
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Scientific calculator
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Put it to work
- Unit 6 problem 4 · No calculator · analytic
- Unit 6 problem 5 · No calculator · analytic
For the teacher
- Ask students to write both a limit and an integral for one context. Check that the variable used in the integrand is a dummy variable and does not change the quantity.
Glossary and deeper reading · 4 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Riemann sum
A Riemann sum approximates accumulation by adding sample heights times subinterval widths. Left, right, and midpoint sums choose different sample positions in each subinterval. For example, a left sum for x² on [0,2] with two equal widths uses heights 0 and 1, giving 1. A definite integral is the limiting value as the largest subinterval width approaches zero when that limit exists.
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Continuity
A function is continuous at an input when its value exists, its finite limit exists, and those two values agree. At a domain endpoint, continuity uses the relevant one-sided approach. For example, x² is continuous at 3 because both its value and its nearby limit are 9. Continuity on an interval requires the appropriate condition at every input in that interval.
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Limit
A limit describes the value approached by outputs as inputs approach a specified target or grow without bound. For an input approaching a finite target, its value at that exact input does not determine the limit. For example, (x²−1)/(x−1) approaches 2 as x approaches 1 even though the original expression is undefined at 1. A finite two-sided limit requires the left and right approaches to agree.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.