Applications of Integration

This lesson covers the practical applications of integration — particularly in kinematics and modelling — as required by the Edexcel A-Level Mathematics specification (9MA0). You need to be able to find displacement from velocity, velocity from acceleration, and apply integration to real-world modelling problems.

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Integration in Kinematics

Kinematics is the study of motion. The three key quantities are:

• Displacement s (position, measured from a reference point)
• Velocity v (rate of change of displacement, v = ds/dt)
• Acceleration a (rate of change of velocity, a = dv/dt)

The Relationships

Differentiation goes "down" this chain:

• s → v = ds/dt
• v → a = dv/dt

Integration goes "up" this chain:

• a → v = ∫ a dt
• v → s = ∫ v dt

To find	You integrate	Formula
Velocity from acceleration	∫ a dt	v = ∫ a dt
Displacement from velocity	∫ v dt	s = ∫ v dt
Distance travelled over [t₁, t₂]	∫ from t₁ to t₂ of	v

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