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IB DP Mathematics: Applications and Interpretation | SL & HL | Calculus (Topic 5)
📋 Learning Intentions
- Define integration as the reverse process of differentiation (anti-differentiation) and state the meaning of the constant of integration \( C \).
- Apply the power rule for integration to find indefinite integrals of polynomial, rational, and exponential functions.
- Find the constant of integration \( C \) by substituting a given boundary condition (initial condition) into the general solution.
- Determine a particular solution to a differential equation of the form \( \frac{dy}{dx} = f(x) \) given a point on the curve.
- Use the GDC to verify anti-derivatives and check particular solutions.
- Interpret the constant of integration in real-world modelling contexts such as kinematics and population growth.
Real-World Introduction
A meteorologist measures the rate at which temperature is changing throughout the day: at 6 am the rate is \( +2 \)°C per hour, rising to \( +5 \)°C per hour by noon, then falling negative in the evening. This rate-of-change function is the derivative of temperature with respect to time. But the meteorologist wants to know the actual temperature at any given hour — not just how fast it is changing.
Recovering the original function from its derivative is exactly what integration does. In kinematics, if acceleration is known, integration gives velocity; integrating velocity gives displacement. In biology, if a population's growth rate is modelled, integration recovers the population size. The process works in every context where a rate of change is measured but the underlying quantity is what matters.
🌍 Why This Matters GPS navigation systems continuously record velocity (rate of change of position). To calculate where a vehicle actually is, the system integrates velocity over time — recovering position from its derivative. Without integration, real-time positioning would be impossible.
Core Concepts & Definitions
Differentiation takes a function and produces its gradient function. Integration reverses this: given a gradient function \( f(x) \), integration finds a function \( F(x) \) such that \( F'(x) = f(x) \). This reverse process is called anti-differentiation.
🔑 Key Definition — Indefinite Integral The indefinite integral of a function \( f(x) \) with respect to \( x \) is written \( \int f(x)\, dx \) and represents the family of all anti-derivatives of \( f(x) \). It is called "indefinite" because no specific numerical limits are placed on the integration. The result always