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Learn through experiments

Approximate accumulation without hiding its error

How do left rectangles, midpoint rectangles and trapezoids differ?

Entering midway? An integral accumulates small contributions. A numerical rule replaces each curved piece with a simpler area.

1 · Try

Start with a concrete question

Integrate x² from zero to a selected endpoint using at most sixteen pieces and compare against the exact integral.

Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.

Go directly to the experiment · Review prerequisite mathematics and return here

2 · Predict

What do you expect?

Which rule underestimates an increasing convex curve, and which overestimates it?

3 · Experiment

Change one thing. Inspect what happens.

  1. Compare the three rules with two pieces on [0,2].
  2. Double piece count while holding the rule fixed; compare signed errors.
  3. Inspect each polygon contribution and sum it independently.
  4. Set endpoint zero, then explain why improving this arithmetic does not validate an empirical model.
Experiment inputs
Inputs
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Numerical area1.75
Exact integral2.66667
Signed approximation error-0.916667
Width per piece0.5

Quadrature pieces under a fixed curve

Horizontal: x. Vertical: x².

001.54.539Piece 1: (0, 0)Piece 1: (0, 0)Piece 1: (0.5, 0)Piece 1: (0.5, 0)Piece 2: (0.5, 0)Piece 2: (0.5, 0.25)Piece 2: (1, 0.25)Piece 2: (1, 0)Piece 3: (1, 0)Piece 3: (1, 1)Piece 3: (1.5, 1)Piece 3: (1.5, 0)Piece 4: (1.5, 0)Piece 4: (1.5, 2.25)Piece 4: (2, 2.25)Piece 4: (2, 0)
  • 1. Exact curve (solid line)
  • 2. Piece 1 (dashed line)
  • 3. Piece 2 (dashed line)
  • 4. Piece 3 (dashed line)
  • 5. Piece 4 (dashed line)

Axes use the displayed ranges.

Read plotted values as a table
Seriesxx²
Exact curve00
Exact curve0.033330.001111
Exact curve0.066670.004444
Exact curve0.10.01
Exact curve0.13330.01778
Exact curve0.16670.02778
Exact curve0.20.04
Exact curve0.23330.05444
Exact curve0.26670.07111
Exact curve0.30.09
Exact curve0.33330.1111
Exact curve0.36670.1344
Exact curve0.40.16
Exact curve0.43330.1878
Exact curve0.46670.2178
Exact curve0.50.25
Exact curve0.53330.2844
Exact curve0.56670.3211
Exact curve0.60.36
Exact curve0.63330.4011
Exact curve0.66670.4444
Exact curve0.70.49
Exact curve0.73330.5378
Exact curve0.76670.5878
Exact curve0.80.64
Exact curve0.83330.6944
Exact curve0.86670.7511
Exact curve0.90.81
Exact curve0.93330.8711
Exact curve0.96670.9344
Exact curve11
Exact curve1.0331.068
Exact curve1.0671.138
Exact curve1.11.21
Exact curve1.1331.284
Exact curve1.1671.361
Exact curve1.21.44
Exact curve1.2331.521
Exact curve1.2671.604
Exact curve1.31.69
Exact curve1.3331.778
Exact curve1.3671.868
Exact curve1.41.96
Exact curve1.4332.054
Exact curve1.4672.151
Exact curve1.52.25
Exact curve1.5332.351
Exact curve1.5672.454
Exact curve1.62.56
Exact curve1.6332.668
Exact curve1.6672.778
Exact curve1.72.89
Exact curve1.7333.004
Exact curve1.7673.121
Exact curve1.83.24
Exact curve1.8333.361
Exact curve1.8673.484
Exact curve1.93.61
Exact curve1.9333.738
Exact curve1.9673.868
Exact curve24
Exact curve2.0334.134
Exact curve2.0674.271
Exact curve2.14.41
Exact curve2.1334.551
Exact curve2.1674.694
Exact curve2.24.84
Exact curve2.2334.988
Exact curve2.2675.138
Exact curve2.35.29
Exact curve2.3335.444
Exact curve2.3675.601
Exact curve2.45.76
Exact curve2.4335.921
Exact curve2.4676.084
Exact curve2.56.25
Exact curve2.5336.418
Exact curve2.5676.588
Exact curve2.66.76
Exact curve2.6336.934
Exact curve2.6677.111
Exact curve2.77.29
Exact curve2.7337.471
Exact curve2.7677.654
Exact curve2.87.84
Exact curve2.8338.028
Exact curve2.8678.218
Exact curve2.98.41
Exact curve2.9338.604
Exact curve2.9678.801
Exact curve39
Piece 100
Piece 100
Piece 10.50
Piece 10.50
Piece 20.50
Piece 20.50.25
Piece 210.25
Piece 210
Piece 310
Piece 311
Piece 31.51
Piece 31.50
Piece 41.50
Piece 41.52.25
Piece 422.25
Piece 420

Follow the calculation

  1. Piece 1: [0.00000,0.50000], width 0.50000, left endpoint height 0.00000, contribution 0.00000.
  2. Piece 2: [0.50000,1.00000], width 0.50000, left endpoint height 0.25000, contribution 0.12500.
  3. Piece 3: [1.00000,1.50000], width 0.50000, left endpoint height 1.00000, contribution 0.50000.
  4. Piece 4: [1.50000,2.00000], width 0.50000, left endpoint height 2.25000, contribution 1.12500.
  5. Exact ∫₀^2 x² dx = 2³/3 = 2.66667; numerical−exact = -0.91667.

Finer pieces reduce this discretization error. That does not test whether x² is a good model of any measured process.

Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.

4 · Understand

Explain the mechanism

Start with the meaning

An integral accumulates small contributions. A numerical rule replaces each curved piece with a simpler area.

Integrate x² from zero to a selected endpoint using at most sixteen pieces and compare against the exact integral.

A worked explanation

A left rectangle uses the function at the interval start. A midpoint rectangle samples its middle. A trapezoid averages endpoint heights.

On [0,2] with two pieces, left rectangles give 1, midpoint rectangles 0.25+2.25=2.5, and trapezoids 0.5+2.5=3. The exact integral is 8/3≈2.6667.

For this smooth curve, refinement reduces discretization error. Model error is separate: a perfectly integrated wrong curve can still give a wrong real-world prediction.

Symbols and relationship

h=(b−a)/n. Left: hΣf(left); midpoint: hΣf(mid); trapezoid: hΣ(f(left)+f(right))/2. Exact ∫₀ᵇx²dx=b³/3.

Use the graph and trace together

The graph and calculations use your current inputs. Read axis labels and units before comparing positions or slopes. Change one input, predict the effect, and use the numerical trace to verify the plotted behavior. A displayed curve is an illustration; the formula and its assumptions determine which conclusions are justified.

Scope and edge cases

  • A smooth quadratic integrand and equal widths; no claim about discontinuities or singular integrals.
  • All polygons share fixed axes; increasing pieces changes numerical work, not the underlying model.

Return to the course and cumulative project

5 · Apply

Try a fresh case

Use two midpoint rectangles to integrate x² from 0 to 4. What is the approximation?

This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.

Take the investigation further

Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.

School investigation

Draw and add the approximation pieces.

College investigation

Derive the error order and test a nonsmooth counterexample.

Workplace investigation

Specify a numerical tolerance and separately document uncertainty in the modeled rate.

6 · Continue

Keep the evidence, choose your next step

Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.