Skip to main content
MathidiaMathematics Academy
Integration
Lesson 40 of 75
Review

Integration modelling

Objective. Interpret accumulated change in mathematical contexts.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Integration modelling interprets a definite or indefinite integral as the total accumulated change of a quantity whose rate of change is known, such as recovering distance travelled from a velocity function or total cost from marginal cost. If v(t) gives velocity, then ∫v(t) dt gives displacement, and a definite integral ∫ₐᵇ v(t) dt gives the displacement between times a and b, while the integral of speed gives total distance travelled. For example, if v(t) = 6t − 2 metres per second, the displacement from t = 0 to t = 3 is ∫₀³ (6t − 2) dt = [3t² − 2t]₀³ = 27 − 6 = 21 metres. Correctly identifying what quantity the integral represents in context is essential to interpreting the final numerical answer.

Worked example

A particle has velocity v(t) = 4t − 8 m/s for t ≥ 0. Use integration modelling to find its displacement from t = 0 to t = 5 seconds.

48005
Bar chart of the values 4, 8, 0, 0, 5
  1. 1Find the antiderivative: s(t) = 2t² − 8t + c, and since displacement is measured from t = 0 let c = 0.
  2. 2Evaluate at t = 5: s(5) = 2(25) − 40 = 50 − 40 = 10.
  3. 3Evaluate at t = 0: s(0) = 0.
  4. 4Displacement = s(5) − s(0) = 10 − 0 = 10 metres.

Answer: 10 metres

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Students often confuse the integral of velocity (displacement) with total distance, not accounting for direction changes when velocity becomes negative. Each interval where velocity is negative must be treated separately and its magnitude added for true distance.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard