Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Calculate a composite trapezoidal estimate.
- Compare an optional Simpson estimate.
- Explain why sparse samples do not prove an error bound.
Before you start
Average two numbers, multiply rates by time, and interpret area under a graph.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A flow sensor records 2, 4, and 5 litres per minute at times 0, 1, and 2 minutes. Estimate the volume entering during those two minutes.
Why this math matters
Real sensors provide discrete readings rather than a perfect formula. Numerical integration fills the gaps with a stated approximation. Different interpolation choices can give different totals from the same measurements. Being clear about that choice is more informative than printing many decimal places and implying the estimate is exact.
Set up the model
A useful answer starts with clear assumptions:
- Readings are equally spaced one minute apart and are exact at those times.
- The trapezoidal model joins consecutive readings by straight lines.
- Simpson's comparison uses a quadratic interpolation; the actual in-between flow is unknown.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Can three readings estimate the total flow?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A flow sensor records 2, 4, and 5 litres per minute at times 0, 1, and 2 minutes. Estimate the volume entering during those two minutes.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Estimate the first minute
V₁ ≈ (2 + 4)/2 × 1 = 3 L
A straight segment between the first two rates has average height three. Multiplying by the one-minute width converts rate to volume.
Estimate the second minute and total
V₂ ≈ (4 + 5)/2 × 1 = 4.5 L; total ≈ 7.5 L
Add the two trapezoid areas. This result is exact for the assumed piecewise-linear interpolation, but not necessarily for the actual flow.
Compare another interpolation
Simpson estimate = (1/3)[2 + 4(4) + 5] = 23/3 ≈ 7.667 L
Simpson's weights integrate the quadratic through the three readings. Its different result highlights uncertainty about the unobserved curve.
The result
The trapezoidal estimate is 7.5 litres. A Simpson estimate from the same samples is approximately 7.667 litres.
The difference between estimates is not a guaranteed error bound. A sharp unobserved spike could defeat both. More frequent readings or justified derivative bounds can support a stronger accuracy assessment.
Common mistakes to catch
- Adding 2 + 4 + 5 treats three endpoint samples as three full time intervals.
- Unequal time spacing cannot use the simple equal-width Simpson formula shown here.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Rates of 1 and 5 L/min are measured two minutes apart. Find the trapezoidal estimate.
Show a hint
Multiply their average by the full interval width.
Reveal answer and explanation
6 litres
(1 + 5)/2 × 2 = 6 litres under straight-line interpolation.
Practice 2
Rates 3, 3, 3 L/min are recorded at 0, 1, 2 minutes. What do both methods estimate?
Show a hint
The interpolated rate is constant.
Reveal answer and explanation
6 litres
Trapezoids give 3 + 3 = 6; Simpson gives (3 + 12 + 3)/3 = 6.
Take the idea with you
Whenever a total comes from sampled rates, record the sampling interval, interpolation method, and limits of the estimate.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Use a derivative to find the largest rectangle
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