This is Chapter 4 – Integration for the Year 12 Mathematical Methods course.

Integration introduces students to the mathematics of accumulation and provides a way of reversing differentiation. While differentiation examines how quickly a quantity is changing, integration can be used to determine how much has accumulated over an interval. Together, these processes form the central framework of calculus.

We begin by exploring the area under a curve. This provides an intuitive starting point for definite integrals and helps students understand why integration is more than simply applying an algebraic rule. Antidifferentiation is then introduced as the process of recovering a family of functions from a known derivative, including the importance of the constant of integration.

The Fundamental Theorem of Calculus brings these ideas together by establishing the connection between an antiderivative and the exact area represented by a definite integral. This is one of the most important results in the course because it unifies the seemingly different ideas of gradients and accumulated area.

Students then develop the standard integration rules needed for polynomial, exponential and trigonometric expressions. Integrals involving functions of the form require students to recognise the effect of an inner linear expression and adjust their working accordingly. Definite integrals are evaluated over stated intervals and interpreted in relation to areas and accumulated quantities.

Integration requires strong differentiation knowledge, since students must often recognise which function would have produced a given derivative. It also demands accurate algebra and careful attention to notation, limits and constants. These techniques are applied further in the following chapter and are widely used in physics, engineering, economics and population modelling.

Subchapters:

• 4A: Area under a curve

• 4B: Antidifferentiation
• 4C: The Fundamental Theorem of Calculus
• 4D: Integration
• 4E: Rules for integration
• 4F: Integrating f(ax+b)
• 4G: Definite integrals

By the end of this chapter, students will be able to find antiderivatives, evaluate definite integrals and explain the fundamental relationship between differentiation, integration and accumulated change.