Accumulation and integration
Change you can see.
Explore limits, derivatives, and accumulation through motion, measurement, and design.
Follow the real situation, derive the mathematics, and test it with interactive graphs and worked practice.
Start exploring ↗Reading your saved completion. Completion is saved on this browser.
Accumulation and integration
An integral grows from many small contributions.
A changing velocity contributes a little displacement over each short time interval. Rectangles make the approximation visible.
Explore the idea ↗Interactive lessonDifferentiation and accumulation undo one another.
Accumulated volume or energy changes at the rate entering at its current endpoint. See the accumulated curve and its slope together.
Explore the idea ↗Interactive lessonInitial data selects an antiderivative.
Two carts can have the same velocity history while starting at different positions. The integration constant carries that missing information.
Explore the idea ↗Interactive lessonDisplacement can cancel while distance accumulates.
A cart reverses direction. Its signed displacement and total distance answer different questions.
Explore the idea ↗Interactive lessonReverse the chain rule with a new variable.
A composed rate law can become simpler when its inner quantity is treated as the integration variable.
Explore the idea ↗