Mathematics · 34 visual lessons

Integration techniques
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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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Unit 1 · 7 lessons

Integration techniques

A time-weighted decaying response combines a polynomial with an exponential. Integration by parts reorganizes this product.

The time-average of a squared oscillation is connected to an integral of sine squared. Reduce the power before integrating.

A circular cross-section produces square roots such as sqrt(a²−x²). An angle parameter makes that geometry simpler.

A uniformly weighted flexible cable hangs in a catenary rather than a parabola. Hyperbolic functions describe its ideal shape.

A rational response with multiple factors can integrate more easily when decomposed into simple reciprocal terms.

A curve can stretch forever while its area remains finite. Compare tail exponents and increasingly distant cutoffs.

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