Integrals add up change
Whenever a quantity accumulates from a rate, integration finds the total change:
∫ₐᵇ (rate) dt = total change
This idea of accumulation appears across science, economics and engineering.
1. Motion
- Displacement = ∫ v(t) dt
- Total distance = ∫ |v(t)| dt
Example: v(t) = t² − 4t + 3 m/s from t = 0 to t = 3.
- Displacement: [t³/3 − 2t² + 3t]₀³ = 9 − 18 + 9 = 0 m (it returned to the start)
- v changes sign at t = 1. Distance: ∫₀¹ v dt + |∫₁³ v dt| = 4/3 + 4/3 = 8/3 m
2. Average value
The average value of f on [a, b]:
f_avg = (1/(b − a)) ∫ₐᵇ f(x) dx
Example: Temperature T(t) = 10 + 6 sin(πt/12) over 12 hours (t in hours).
∫₀¹² 6 sin(πt/12) dt = 6 × (12/π) × [−cos(πt/12)]₀¹² = (72/π)(2) = 144/π
Average = (1/12)(120 + 144/π) ≈ 10 + 3.82 = 13.8 °C
3. Volumes of revolution
Rotating a region around the x-axis creates a solid. Each thin slice is a disc of area π[f(x)]².
V = π ∫ₐᵇ [f(x)]² dx
This is a volume of revolution.
Example: Rotate y = √x from 0 to 4 around the x-axis.
V = π ∫₀⁴ x dx = π[x²/2]₀⁴ = 8π ≈ 25.1 units³
Check with a cone: Rotating y = (r/h)x from 0 to h gives V = (1/3)πr²h ✓.
4. Economics
- Total cost from marginal cost: C(b) − C(a) = ∫ₐᵇ C′(x) dx
- Total revenue from a revenue rate
Example: Marginal cost C′(x) = 50 − 0.2x dollars per unit. Cost to increase production from 100 to 150 units?
∫₁₀₀¹⁵⁰ (50 − 0.2x) dx = [50x − 0.1x²]₁₀₀¹⁵⁰ = (7,500 − 2,250) − (5,000 − 1,000) = $1,250
5. Flow and growth
- Water entering a reservoir from an inflow rate
- Population change from a growth rate
- Total rainfall from an intensity graph
6. Physics: work
Work = ∫ F(x) dx. Stretching a spring with F = kx from 0 to d: W = ½kd².
Worked examples
Example 1: Oil leaks at r(t) = 4e^(−0.5t) L/h. Total leaked in the first 2 hours? ∫₀² 4e^(−0.5t) dt = 8(1 − e⁻¹) ≈ 5.06 L.
Example 2: Average value of f(x) = x² on [0, 3]? (1/3)(9) = 3.
Example 3: Volume when y = 2 (from 0 to 5) rotates around the x-axis? π(4)(5) = 20π (a cylinder ✓).
Common mistakes
- Confusing displacement and total distance.
- Forgetting π or squaring f(x) in volume problems.
- Forgetting to divide by (b − a) for average value.
Try it yourself
- v(t) = 2t + 1. Displacement from t = 0 to t = 3?
- Average value of f(x) = 4x on [0, 2]?
- Rotate y = x from 0 to 3 around the x-axis. Volume?
- A rate of 3t² L/min flows in for 2 minutes. Total?
Answers: 1) 12 2) 4 3) 9π 4) 8 L
Quick recap
- Integration turns rates into total change through accumulation.
- Average value = integral ÷ interval length.
- Volume of revolution: π ∫ [f(x)]² dx; integrate velocity for displacement.