Integration and accumulation - Mathematics: Analysis & Approaches IB Study Notes

Integration is the reverse process of differentiation, often called antidifferentiation. While differentiation finds rates of change, integration finds the accumulation of quantities over intervals.

Indefinite Integration produces a family of functions: ∫f(x)dx = F(x) + C, where F'(x) = f(x) and C is the constant of integration. The constant C represents an infinite family of antiderivatives, each differing by a vertical translation.

Definite Integration calculates the exact accumulated value between bounds: ∫ₐᵇf(x)dx = F(b) - F(a), using the Fundamental Theorem of Calculus. This theorem connects differentiation and integration, stating that integration "undoes" differentiation.

Key Integration Rules:

• Power Rule: ∫xⁿdx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1
• Special case: ∫(1/x)dx = ln|x| + C
• Exponential: ∫eˣdx = eˣ + C and ∫aˣdx = aˣ/ln(a) + C
• Trigonometric: ∫sin(x)dx = -cos(x) + C; ∫cos(x)dx = sin(x) + C

Accumulation represents the total change in a quantity. When velocity v(t) is integrated over time, you obtain displacement: s = ∫v(t)dt. The definite integral ∫ₐᵇf(x)dx geometrically represents the signed area between the curve f(x) and the x-axis from x=a to x=b (areas below the axis are negative).

Cambridge Note: Always include the constant of integration C for indefinite integrals—omitting it costs marks. For definite integrals, show substitution of bounds clearly using square bracket notation: [F(x)]ₐᵇ = F(b) - F(a).

Integration models accumulation in real-world contexts. Think of a water tank: if you know the rate water flows in (litres per minute), integration tells you the total volume accumulated over time. Differentiation would reverse this—starting with volume to find flow rate.

Real-World Applications:

1. Distance from Velocity: A car's speedometer shows instantaneous velocity v(t). To find total distance traveled from t=0 to t=5 seconds, integrate: Distance = ∫₀⁵v(t)dt. If v(t) = 20 + 3t m/s, then Distance = [20t + 1.5t²]₀⁵ = 100 + 37.5 = 137.5 metres.

2. Economics—Total Revenue: If marginal revenue MR(q) = 50 - 0.4q (revenue per additional unit sold), total revenue from selling 100 units is R = ∫₀¹⁰⁰(50 - 0.4q)dq = [50q - 0.2q²]₀¹⁰⁰ = 5000 - 2000 = $3000.

3. Population Growth: If population grows at rate P'(t) = 1000e⁰·⁰²ᵗ people/year, the population increase over 10 years is ∫₀¹⁰1000e⁰·⁰²ᵗdt = [50000e⁰·⁰²ᵗ]₀¹⁰ = 50000(e⁰·² - 1) ≈ 11,070 people.

Analogy: Think of integration as collecting rainwater. The rainfall rate (mm/hour) is like the function f(x). Integration accumulates this rate over time to give total rainfall collected—the area under the "rate curve." Just as you need a collection interval (bucket placement times a to b), definite integrals need bounds.

Connection: Signed area matters—if velocity is negative (moving backward), the displacement integral reflects this directional change, crucial for physics applications.