Calculus and numerical methods

Integration starts with a simple question: how much area is under the curve?

This prototype compares rectangle and trapezium approximations with exact area, helping learners see definite integration as a limiting process.

The aim is to connect the visual idea of area with numerical methods, error and exact integration.

Integration lab

Approximate area under a curve

Compare numerical area approximations with the exact integral. More strips usually improve the estimate, but the method and shape of the curve matter.

Approximation

18.9355

Exact integral

18.8542

Absolute error

0.0814

Strip width

0.625

Controls

Function

Method

Number of strips: 8

Lower limit: 0.5

Upper limit: 5.5

f(x) = 0.25x^2 + 1

A steadily increasing curve where left rectangles underestimate the area.

Current method: trapezium rule.

What this shows

Definite integration can be understood as the limit of better and better area approximations.

Rectangles and trapezia are numerical methods: they estimate the area without needing the exact antiderivative.

The error depends on the number of strips, the method and the shape of the curve.

Guided tasks

Increase the number of strips. Does the error always decrease smoothly?

Compare left rectangles with the trapezium rule on the quadratic. Which is closer?

Change the limits. Why does the same method behave differently on different parts of a curve?

Joy in the process

The point is not just to finish. It is to notice, test and return.

These tools are invitations to explore. A good mistake, a surprising pattern or a question you cannot yet answer is part of the work, not a failure of it.

The challenge is deliberate: the site should support thinking, not remove the need for it.

Before changing a setting, pause and predict what you think will happen.

Change one thing at a time. What stayed the same, and what changed?

Try to create a surprising case, a broken case, or a beautiful pattern.

Ask what this connects to outside the page: maps, movement, nature, systems or decisions.

Reset, then try again with a new question in mind.

Future extensions

This can grow into a broader integration and numerical methods lab.

Add right rectangles and Simpson's rule.

Add animated convergence as the number of strips increases.

Add area between two curves.

Add exam-style prompts for trapezium rule and definite integration.