Unit 1 · Lesson 7 of 7

Estimate an integral when no elementary formula is handy.

A measured rate can be sampled at discrete times. Trapezoids or parabolic panels turn those observations into an accumulated quantity.

Change the input. Follow the mathematics.

What you'll be able to do

Set up the model and calculate its quadrature 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 even panel count n? Predict the direction, then explain the response.

A model you can changeMove one control at a time.

Read the model

T_n=h\left[\frac{f_0+f_n}{2}+\sum_{i=1}^{n-1}f_i\right],\quad S_n=\frac h3[f_0+4f_1+2f_2+\cdots+f_n]

Integrate exp(−x²) on [0,2]. Simpson uses even n. A high-resolution Simpson value provides a numerical reference, not an elementary exact antiderivative.

Sampled rate

Even panel count n6

quadrature estimate

h=\frac2{6},\quad S_6=0.882032,\quad I_{ref}\approx0.882081

Smoothness, panel width, and rule order affect accuracy. More panels are not a substitute for understanding discontinuities or singular endpoints.

Assumptions, units, and limits

Integrate exp(−x²) on [0,2]. Simpson uses even n. A high-resolution Simpson value provides a numerical reference, not an elementary exact antiderivative.

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

From the situation to an equation

Trapezoid weights are one half at endpoints and one inside. Simpson weights alternate four and two between endpoint ones, and require an even number of panels. Error behavior depends on smoothness and derivatives of the integrand.

T_n=h\left[\frac{f_0+f_n}{2}+\sum_{i=1}^{n-1}f_i\right],\quad S_n=\frac h3[f_0+4f_1+2f_2+\cdots+f_n]

Keep the reasoning visible

h=(b-a)/n

Smoothness, panel width, and rule order affect accuracy. More panels are not a substitute for understanding discontinuities or singular endpoints.

What makes this model useful?

Integrate exp(−x²) on [0,2]. Simpson uses even n. A high-resolution Simpson value provides a numerical reference, not an elementary exact antiderivative.

The reference displayed here is itself a high-resolution numerical approximation. More samples do not automatically repair an unmodeled discontinuity.

A worked example

Calculate the starting scenario

Use even panel count n = 6 ; rule: 0 trapezoid, 1 simpson = 1 .
$h=\frac2{6},\quad S_6=0.882032,\quad I_{ref}\approx0.882081$
Smoothness, panel width, and rule order affect accuracy. More panels are not a substitute for understanding discontinuities or singular endpoints. The checked quantity is quadrature 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 quadrature estimate using even panel count n 6 , rule: 0 trapezoid, 1 simpson 1 . State the meaning of its sign and units.

Checked quantity: quadrature estimate · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Explore the visual solution

Read the model

T_n=h\left[\frac{f_0+f_n}{2}+\sum_{i=1}^{n-1}f_i\right],\quad S_n=\frac h3[f_0+4f_1+2f_2+\cdots+f_n]

Integrate exp(−x²) on [0,2]. Simpson uses even n. A high-resolution Simpson value provides a numerical reference, not an elementary exact antiderivative.

Sampled rate

Even panel count n6

quadrature estimate

h=\frac2{6},\quad S_6=0.882032,\quad I_{ref}\approx0.882081

Smoothness, panel width, and rule order affect accuracy. More panels are not a substitute for understanding discontinuities or singular endpoints.

Assumptions, units, and limits

Integrate exp(−x²) on [0,2]. Simpson uses even n. A high-resolution Simpson value provides a numerical reference, not an elementary exact antiderivative.

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.

T_n=h\left[\frac{f_0+f_n}{2}+\sum_{i=1}^{n-1}f_i\right],\quad S_n=\frac h3[f_0+4f_1+2f_2+\cdots+f_n]

Calculate and explain

Apply the idea to a new scenario

Use 10 trapezoid panels to integrate exp(−x²) from 0 to 2.

Checked quantity: quadrature estimate · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Explore the visual solution

Read the model

T_n=h\left[\frac{f_0+f_n}{2}+\sum_{i=1}^{n-1}f_i\right],\quad S_n=\frac h3[f_0+4f_1+2f_2+\cdots+f_n]

Integrate exp(−x²) on [0,2]. Simpson uses even n. A high-resolution Simpson value provides a numerical reference, not an elementary exact antiderivative.

Sampled rate

Even panel count n10

quadrature estimate

h=\frac2{10},\quad T_10=0.881839,\quad I_{ref}\approx0.882081

Smoothness, panel width, and rule order affect accuracy. More panels are not a substitute for understanding discontinuities or singular endpoints.

Assumptions, units, and limits

Integrate exp(−x²) on [0,2]. Simpson uses even n. A high-resolution Simpson value provides a numerical reference, not an elementary exact antiderivative.

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.

T_n=h\left[\frac{f_0+f_n}{2}+\sum_{i=1}^{n-1}f_i\right],\quad S_n=\frac h3[f_0+4f_1+2f_2+\cdots+f_n]

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

T_n=h\left[\frac{f_0+f_n}{2}+\sum_{i=1}^{n-1}f_i\right],\quad S_n=\frac h3[f_0+4f_1+2f_2+\cdots+f_n]

Integrate exp(−x²) on [0,2]. Simpson uses even n. A high-resolution Simpson value provides a numerical reference, not an elementary exact antiderivative.

Sampled rate

Even panel count n6

quadrature estimate

h=\frac2{6},\quad S_6=0.882032,\quad I_{ref}\approx0.882081

Smoothness, panel width, and rule order affect accuracy. More panels are not a substitute for understanding discontinuities or singular endpoints.

Assumptions, units, and limits

Integrate exp(−x²) on [0,2]. Simpson uses even n. A high-resolution Simpson value provides a numerical reference, not an elementary exact antiderivative.

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

The reasoning. Trapezoid weights are one half at endpoints and one inside. Simpson weights alternate four and two between endpoint ones, and require an even number of panels. Error behavior depends on smoothness and derivatives of the integrand.

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.

Completion is saved on this browser. This is separate from account practice progress.