Integrals in the physical world
Change you can see.
Explore integration, differential equations, series, and geometry through physical models.
Follow the real situation, derive the mathematics, and test it with interactive graphs and worked practice.
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Integrals in the physical world
Probability is area beneath a density.
Waiting times can be modeled by a density curve. The chance of an event before a deadline is accumulated area, not curve height.
Explore the idea ↗Interactive lessonCurve length adds tiny diagonal segments.
A road or cable follows a curve rather than a straight horizontal line. Its length must include both horizontal and vertical change.
Explore the idea ↗Interactive lessonRotation sweeps circumference along a slant length.
A cone's side area comes from rings with circumference set by radius and slant thickness set by the curve.
Explore the idea ↗Interactive lessonPressure builds force one submerged strip at a time.
A deeper gate experiences more pressure and a force that acts below its center. Change gate height, width, and top depth to see all three effects.
Explore the idea ↗Interactive lessonThe balance point is a weighted average.
A rod can be heavier at one end even with a uniform shape. Its balance point depends on mass distribution.
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