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A course in building mathematical models

Calculus II.
See the whole system.

How much force does water exert? How will a population grow? How accurate is an approximation? Turn small relationships into predictions about the whole.

Before the equations,
a question to play with.

Lower the gate. Change its height. Does twice the height mean twice the force?

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FIG. II / THE WEIGHT OF WATER
Water surface1.0 m2.0 m tallLonger arrows → greater pressure
F=ρgb∫0H(d+y) dy=78.48 kNF=\rho gb\int_0^H(d+y)\,dy=78.48\,\mathrm{kN}

Force acts 1.17 m below the gate’s top (red dot). Fresh water; width 2 m; gravity 9.81 m/s².

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Why this course matters

A small relationship.
A powerful prediction.

Designing a water gate. Predicting cooling. Estimating a curved path or a population’s future. Calculus II turns local information into models of geometry, force, and evolving systems.

You’ll learn how to choose a method, state its assumptions, and judge its accuracy. The mathematics stays visible alongside the physical meaning.

What we’ll investigate

Three ways to build a model.

Choose a question

Deeper water. A larger local load.

Pressure increases with depth, so a submerged gate does not experience one uniform pressure. Slice the gate into narrow strips, calculate the force on each strip, and integrate to find the total load and where it acts.

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Your learning path · 34 lessons

A little structure.
A world to discover.

Build your integration toolkit, apply it to the physical world, then explore evolving systems, infinite sums, and spatial geometry.

Each lesson pairs visual exploration with mathematical derivations and three practice questions. Start at the beginning or revisit the idea you need.

Begin the course ↗

Revisit Calculus I ↗

IIntegration techniques+

Choose integration techniques and compare numerical estimates.

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IIIntegrals in the physical world+

Build integral models for probability, geometry, and fluid loads.

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IIIDifferential equations and models+

Predict evolving states with differential equations and initial conditions.

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IVSequences and infinite series+

Test convergence and use series with meaningful error estimates.

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VCurves and spatial geometry+

Describe paths, directions, work, and torque with curves and vectors.

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