8.5MechanicsStretch

Integrating Vectors

Integration reverses differentiation for vectors too. From an acceleration vector you integrate to velocity, and again to position — using initial conditions to fix the vector constants.

24 min Further Kinematics

Edexcel A Level Maths: 8.5 Integrating VectorsCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Integrate a vector component-wise
Find velocity from acceleration
Find position from velocity
Use initial conditions for the constant vector

Component-wise integration

Integrate each component and add a constant $vector$ . $v = \int a d t + c$ , with $c$ found from the velocity at a known time.

v = \int a d t + c

Integrate each component; add a constant vector.

\int 2 i d t = 2 t i + c

2At

t = 0

v = 3 j

, so

c = 3 j

Answer

v = 2 t i + 3 j

Tip — The constant of integration is a VECTOR — find it from the initial velocity or position.

Formula recap

v = \int a d t + c

Velocity from acceleration.

r = \int v d t + c

Position from velocity.

Common mistakes to avoid

Adding a scalar constant of integration.

The constant is a vector (it has i and j parts).

Skipping the initial conditions.

Use the given velocity/position to find the constant vector.

Key takeaways

Integrate each component, add a constant vector.
v = ∫a dt + c; r = ∫v dt + c.
Use initial conditions to find the constant vector.

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