riaology
Unit 4 · Lesson 1 of 5

An integral grows from many small contributions.

A changing velocity contributes a little displacement over each short time interval. Rectangles make the approximation visible.

Change the input. Follow the mathematics.

What you'll be able to do
  • Set up the model and calculate its rectangle-sum estimate.
  • Connect the equation to the changing visual.
  • Explain the assumptions and identify a misleading interpretation.
Predict before exploring

What will change if you increase rectangle count? Predict the direction, then explain the response.

A model you can changeMove one control at a time.
Read the model∑i=1nf(ti∗)Δt→∫02t2 dt=83\sum_{i=1}^nf(t_i^*)\Delta t\to\int_0^2t^2\,dt=\frac83

Integrating velocity t² m/s over 0–2 s. Three sampling rules; rectangle heights are visible in the diagram.

Time (s) versus Rate (m/s). Velocity. Selected rectangle-sum estimate: 2.03704 m. For this increasing function, left endpoints underestimate and right endpoints overestimate. Midpoints converge faster here.002142Rate (m/s)Time (s)
Velocity

rectangle-sum estimateS6=2.03704 m,exact=83 mS_6=2.03704\,\mathrm m,\quad\text{exact}=\frac83\,\mathrm m

For this increasing function, left endpoints underestimate and right endpoints overestimate. Midpoints converge faster here.

Assumptions, units, and limits

Integrating velocity t² m/s over 0–2 s. Three sampling rules; rectangle heights are visible in the diagram.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

01

From the situation to an equation

Partition the interval, choose a sampling point in each subinterval, multiply rate by duration, and sum. Taking a refining-partition limit defines the definite integral. Sample choice affects finite error even though the limit can agree.

∑i=1nf(ti∗)Δt→∫02t2 dt=83\sum_{i=1}^nf(t_i^*)\Delta t\to\int_0^2t^2\,dt=\frac83
02

Keep the reasoning visible

Δs≈∑iv(ti∗)Δt\Delta s\approx\sum_i v(t_i^*)\Delta t

For this increasing function, left endpoints underestimate and right endpoints overestimate. Midpoints converge faster here.

03

What makes this model useful?

Integrating velocity t² m/s over 0–2 s. Three sampling rules; rectangle heights are visible in the diagram.

The rate-height plot and accumulated displacement have different units. Rectangles must include their widths.

A worked example

Calculate the starting scenario

  1. Use rectangle count = 6 ; rule: 0 left, 1 midpoint, 2 right = 0 .
  2. S6=2.03704 m,exact=83 mS_6=2.03704\,\mathrm m,\quad\text{exact}=\frac83\,\mathrm m
  3. For this increasing function, left endpoints underestimate and right endpoints overestimate. Midpoints converge faster here. The checked quantity is rectangle-sum estimate.
Make the idea your own

Calculate. Compare. Explain.

Numeric checks allow 1% rounding error. Values near zero use a small absolute tolerance. Work through the setup before opening a solution.

Calculate and explain

Rebuild the starting calculation

Calculate the rectangle-sum estimate using rectangle count 6 , rule: 0 left, 1 midpoint, 2 right 0 . State the meaning of its sign and units.

Checked quantity: rectangle-sum estimate · m. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Partition the interval, choose a sampling point in each subinterval, multiply rate by duration, and sum. Taking a refining-partition limit defines the definite integral. Sample choice affects finite error even though the limit can agree.

Explore the visual solution
Read the model∑i=1nf(ti∗)Δt→∫02t2 dt=83\sum_{i=1}^nf(t_i^*)\Delta t\to\int_0^2t^2\,dt=\frac83

Integrating velocity t² m/s over 0–2 s. Three sampling rules; rectangle heights are visible in the diagram.

Time (s) versus Rate (m/s). Velocity. Selected rectangle-sum estimate: 2.03704 m. For this increasing function, left endpoints underestimate and right endpoints overestimate. Midpoints converge faster here.002142Rate (m/s)Time (s)
Velocity

rectangle-sum estimateS6=2.03704 m,exact=83 mS_6=2.03704\,\mathrm m,\quad\text{exact}=\frac83\,\mathrm m

For this increasing function, left endpoints underestimate and right endpoints overestimate. Midpoints converge faster here.

Assumptions, units, and limits

Integrating velocity t² m/s over 0–2 s. Three sampling rules; rectangle heights are visible in the diagram.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

Partition the interval, choose a sampling point in each subinterval, multiply rate by duration, and sum. Taking a refining-partition limit defines the definite integral. Sample choice affects finite error even though the limit can agree.

∑i=1nf(ti∗)Δt→∫02t2 dt=83\sum_{i=1}^nf(t_i^*)\Delta t\to\int_0^2t^2\,dt=\frac83
Calculate and explain

Apply the idea to a new scenario

Use 10 right-endpoint rectangles to integrate t² from 0 to 2 s. Find the approximate displacement.

Checked quantity: rectangle-sum estimate · m. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Partition the interval, choose a sampling point in each subinterval, multiply rate by duration, and sum. Taking a refining-partition limit defines the definite integral. Sample choice affects finite error even though the limit can agree.

Explore the visual solution
Read the model∑i=1nf(ti∗)Δt→∫02t2 dt=83\sum_{i=1}^nf(t_i^*)\Delta t\to\int_0^2t^2\,dt=\frac83

Integrating velocity t² m/s over 0–2 s. Three sampling rules; rectangle heights are visible in the diagram.

Time (s) versus Rate (m/s). Velocity. Selected rectangle-sum estimate: 3.08 m. For this increasing function, left endpoints underestimate and right endpoints overestimate. Midpoints converge faster here.002142Rate (m/s)Time (s)
Velocity

rectangle-sum estimateS10=3.08 m,exact=83 mS_10=3.08\,\mathrm m,\quad\text{exact}=\frac83\,\mathrm m

For this increasing function, left endpoints underestimate and right endpoints overestimate. Midpoints converge faster here.

Assumptions, units, and limits

Integrating velocity t² m/s over 0–2 s. Three sampling rules; rectangle heights are visible in the diagram.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

Partition the interval, choose a sampling point in each subinterval, multiply rate by duration, and sum. Taking a refining-partition limit defines the definite integral. Sample choice affects finite error even though the limit can agree.

∑i=1nf(ti∗)Δt→∫02t2 dt=83\sum_{i=1}^nf(t_i^*)\Delta t\to\int_0^2t^2\,dt=\frac83
Challenge the interpretation

Which explanation survives the model?

Choose the statement you can justify, then use the visual to explain why the others fail.

Test it with the visual
Read the model∑i=1nf(ti∗)Δt→∫02t2 dt=83\sum_{i=1}^nf(t_i^*)\Delta t\to\int_0^2t^2\,dt=\frac83

Integrating velocity t² m/s over 0–2 s. Three sampling rules; rectangle heights are visible in the diagram.

Time (s) versus Rate (m/s). Velocity. Selected rectangle-sum estimate: 2.03704 m. For this increasing function, left endpoints underestimate and right endpoints overestimate. Midpoints converge faster here.002142Rate (m/s)Time (s)
Velocity

rectangle-sum estimateS6=2.03704 m,exact=83 mS_6=2.03704\,\mathrm m,\quad\text{exact}=\frac83\,\mathrm m

For this increasing function, left endpoints underestimate and right endpoints overestimate. Midpoints converge faster here.

Assumptions, units, and limits

Integrating velocity t² m/s over 0–2 s. Three sampling rules; rectangle heights are visible in the diagram.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The reasoning. Partition the interval, choose a sampling point in each subinterval, multiply rate by duration, and sum. Taking a refining-partition limit defines the definite integral. Sample choice affects finite error even though the limit can agree.

Your learning path

Can you explain the connection?

Mark complete when you can set up the calculation and explain what the visual is showing. Opening a page never completes it automatically.

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