How do rectangles approximate accumulation?
You will be able to: Compute left, right and midpoint sums and identify their sample points.
How do rectangles approximate accumulation?
A tap changes speed, but you only record one flow reading during each minute. Treating that reading as constant for the whole minute gives a rectangle estimate. Which reading you choose matters.
A useful starting point: Why can equal positive and negative areas give zero change? →
Words and symbols before equations
- Partition
- A division of the interval into smaller subintervals.
- Sample point
- Where the function height is measured for one rectangle.
- Left/right sum
- Uses the left/right endpoint of each subinterval.
- Midpoint sum
- Uses the center of each subinterval.
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 f(x)=x² on [0,2] with two equal pieces, each width is 1. Left heights are f(0)=0 and f(1)=1, giving L₂=1.
Right heights are f(1)=1 and f(2)=4, giving R₂=5. Midpoint heights are f(0.5)=0.25 and f(1.5)=2.25, giving M₂=2.5.
Every term is height times its own width. The exact integral is 8/3, about 2.667; none of these two-rectangle estimates is exact.
Increasing the number of rectangles reduces their widths. For this continuous function the sums approach the integral, but a finite rectangle picture remains an approximation.
A worked example, step by step
Approximate the integral of f(x)=x+1 on [0,4] with two left rectangles and two midpoint rectangles.
- The width is (4−0)/2=2.
- Left sample inputs are 0 and 2; heights are 1 and 3.
- L₂=2(1+3)=8.
- Midpoints are 1 and 3, with heights 2 and 4; M₂=2(2+4)=12, exact here because f is linear.
Do not use all n+1 partition endpoints as n rectangle heights. Each of the n subintervals gets exactly one sample.
For two rectangles on [0,2], are the midpoint inputs 0 and 2?
Compare with an explanation
No. They are 0.5 and 1.5; each lies at a subinterval center.
Predict. Change one thing. Explain.
Keep f=x² on [0,2]. Compare left, right and midpoint with n=2, then increase n. Explain each estimate using the shaded rectangles and their sample positions.
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 questionFor f=x² on [0,2], calculate left and right sums with four equal intervals and compare with 8/3.
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Compare with the answer and four-point rubric
- 1 point: Δx=0.5.
- 1 point: L₄=0.5(0+0.25+1+2.25)=1.75.
- 1 point: R₄=0.5(0.25+1+2.25+4)=3.75.
- 1 point: 1.75<8/3<3.75, consistent with f increasing.
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 sets the width?
The subinterval endpoints; for equal intervals it is (b−a)/n.
RECALL 2What sets a rectangle height?
The function value at its chosen sample point.
RECALL 3Is a finite sum the exact integral?
Usually not; it is an approximation unless special structure makes it exact.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do rectangles approximate accumulation?
- Equal width Δx=(b−a)/n.
- Left input a+(i−1)Δx; right a+iΔx; midpoint a+(i−1/2)Δx.
- Sum height × width.
Remember: Do not use all n+1 partition endpoints as n rectangle heights. Each of the n subintervals gets exactly one sample.
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 BC Unit 6 · Objectives LIM-5.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 6.2, LIM-5.A. CED effective Fall 2020 and Fall 2026 clarifications checked September 17, 2026. BC scope includes all fourteen topics, including integration by parts (6.11), nonrepeating linear partial fractions (6.12) and improper integrals (6.13). Topic 6.14 integrates the BC toolkit 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. The BC extension teaches parts from the product rule, decomposition with distinct linear factors and independent improper limits. Finite truncations do not certify convergence. Shared foundation lessons are maintained with AB; BC extensions and method selection are authored for this course.
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 BC 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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