Maths That Models Change
Integration & Accumulation
14 min · 2 worked examplesStudent plan
Integration adds up a changing quantity over time or distance. It is how you get energy from power, and charge from current.
After this lesson you can
- Read an accumulated quantity as the area under a curve
- Integrate a simple polynomial
- Work out charge from a current-time graph
The reverse of differentiation
If differentiation asks how fast something is changing, integration asks how much has accumulated.
The power rule runs backwards: the integral of xⁿ is xⁿ⁺¹/(n+1). Raise the power by one, then divide by the new power. So 3x² integrates back to x³, as it should.
An indefinite integral carries a constant of integration, "+ C", because any constant differentiates to zero and is therefore invisible to differentiation. Physically that constant is your starting value, the charge already on the capacitor, the distance already travelled. Forgetting it is the same error as ignoring the intercept on a sensor line.
In the full lesson
- The reverse of differentiation
- Area under a curve
- RMS, properly
This lesson comes with the Student plan. The first module of Maths for Engineers is free.