How do we translate between a sum limit and an integral?
You will be able to: Identify width, interval and sample input in both directions.
How do we translate between a sum limit and an integral?
A camera samples a two-minute event at equally spaced times. The labels tell you the starting time, how far apart the samples are, and which sample belongs to each frame. A Riemann sum carries the same information.
A useful starting point: What does every part of a definite integral mean? →
Words and symbols before equations
- n
- Number of equal subintervals.
- i
- Index running from 1 through n.
- Δx
- Equal width (b−a)/n.
- xᵢ
- Right sample input a+iΔx.
What this picture assumes
Original model; readouts are rounded. f=x² on [0,2], with equal intervals; exact integral 8/3. Positive-width shapes show numerical contributions. Dimensionless x and f; finite sums are estimates.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- n=2; Δx=1; estimate 1; exact 8/3≈2.66667; estimate minus exact=-1.66667.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
For ∫₁³ x² dx, the interval length is 2, so Δx=2/n. The right sample is xᵢ=1+2i/n. The corresponding limit is lim as n→∞ of Σ from i=1 to n of (1+2i/n)²(2/n).
Read the width outside the function separately from the sample inside it. The factor 2/n is the width; the expression 1+2i/n is the input to the square function.
Conversely, lim Σ [4+(3i/n)²](3/n) can be read as ∫₀³ (4+x²) dx. Here the starting input is 0, interval length is 3 and right samples are 3i/n.
An equivalent integral may be obtained by a change of variable. A representation need not be unique, but every component of the proposed sum must match consistently.
A worked example, step by step
Write a right-sum limit for ∫₂⁵ (x+1) dx.
- Interval length is 5−2=3, so Δx=3/n.
- Right samples are 2+3i/n.
- Function heights are (2+3i/n)+1=3+3i/n.
- Use lim as n→∞ of Σᵢ₌₁ⁿ (3+3i/n)(3/n); the final sample is xₙ=5.
Do not confuse an added constant inside the function with the interval starting point. First identify the sampled input and width consistently.
Can two different-looking sum limits represent the same integral?
Compare with an explanation
Yes. Left, right and midpoint samples converge to the same value for a continuous function.
Predict. Change one thing. Explain.
For the model x² on [0,2], name the width 2/n and right sample 2i/n. Compare n=2 and n=4, then write the same integral with midpoint samples.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
n=2; Δx=1; estimate 1; exact 8/3≈2.66667; estimate minus exact=-1.66667.
Original model; readouts are rounded. f=x² on [0,2], with equal intervals; exact integral 8/3. Positive-width shapes show numerical contributions. Dimensionless x and f; finite sums are estimates.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the rate units, signed contributions, theorem conditions, domain or antiderivative check. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionTranslate lim Σᵢ₌₁ⁿ (2+4i/n)³(4/n) into an integral, identifying each piece and evaluating it.
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Compare with the answer and four-point rubric
- 1 point: Width 4/n implies total interval length 4.
- 1 point: Sample 2+4i/n begins from 2 and ends at 6.
- 1 point: The integral is ∫₂⁶ x³ dx.
- 1 point: An antiderivative is x⁴/4, giving (1296−16)/4=320.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What check locates the right endpoint?
Insert i=n into the sample input.
RECALL 2Where is Δx in the sum?
It multiplies each sampled function height.
RECALL 3Do we take a limit for a finite estimate?
No; a finite sum estimates, while its refining limit defines the integral.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do we translate between a sum limit and an integral?
- Right-sum form: lim Σ f(a+i(b−a)/n)(b−a)/n.
- Check xₙ=b and nΔx=b−a.
Remember: Do not confuse an added constant inside the function with the interval starting point. First identify the sampled input and width consistently.
Conditions: Original model; readouts are rounded. f=x² on [0,2], with equal intervals; exact integral 8/3. Positive-width shapes show numerical contributions. Dimensionless x and f; finite sums are estimates.
Refresh Kid · AP Calculus AB Unit 6 · Objectives LIM-5.B, LIM-5.C · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 6.3, LIM-5.B, LIM-5.C. CED effective Fall 2020 and Fall 2026 clarifications checked September 17, 2026. AB scope includes 6.1–6.10 and 6.14. Topics 6.11–6.13 are BC-only; the numbering gap is intentional. Topic 6.14 consolidates the AB antidifferentiation objectives and Skill 1.C. Focused lesson titles, examples and questions are original Refresh Kid material.
Distinguish signed accumulation, unsigned area and initial amount. Error direction needs interval-wide shape information. FTC hypotheses, variable-bound chain factors, integration constants, transformed bounds and domain restrictions are explicit. Bounded jumps are treated separately from differentiability of accumulation. No improper-integral evaluation, integration by parts or partial-fraction decomposition is taught in this AB unit.
The Organic Chemistry Tutor Fundamental Theorem of Calculus Part 1 video title, creator and description were checked; the full video was not reviewed. Khan Academy’s unit destination was checked, but its JavaScript lesson contents were not fully readable by the research tool. OpenStax Sections 5.2, 5.3 and 5.5 were consulted for conceptual cross-checking. No provider scripts, questions, diagrams or artwork were copied. Refresh Kid is not affiliated with or endorsed by these providers.
GitHub’s 3D website collection informed optional spatial inspection and camera controls. The original tank geometry uses existing self-hosted Three.js with its MIT license. Its 2×2 dm base and water depth use the same volume as the rate calculation: 1 dm³=1 L. The initial water is blue and newly accumulated inflow is teal. The rate graph is an abstract amount calculation; it is not the physical shape of water. Camera rotation changes only the view. Full 2D graphs, readouts and explanations remain available without WebGL.
Independent teacher review and observation of students remain pending. Technical checks do not certify mathematical accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus AB questions and scoring guides. The archive spans multiple units; no entire exam question is assigned here.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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