This is Chapter 5 – Applications of Integration for the Year 12 Mathematical Methods course.

This chapter takes the integration techniques developed in the previous chapter and applies them to geometric, kinematic and modelling problems. The emphasis shifts from evaluating an integral in isolation to deciding what should be integrated, over which interval and how the result should be interpreted in context.

We begin with area under a curve and then extend this to the area between two functions. Students must identify relevant intersections, determine which function lies above the other and construct an appropriate definite integral. A correct diagram or graph can be extremely useful in preventing sign and order errors.

Kinematics provides an important application of integration. Students move between acceleration, velocity and displacement, using antiderivatives and initial conditions to recover unknown functions. These questions reinforce the relationship between differentiation and integration and require careful attention to units and physical meaning.

Problem solving by integration brings several skills together. Students may need to form a model from written information, select suitable limits, evaluate an integral and explain what the resulting quantity represents. These questions are often multi-step and reward clear planning rather than immediate calculation. Students should also check whether a signed integral or a positive geometric area is required, since this distinction can change how the calculation is set up and interpreted.

Applications of integration appear throughout physics, engineering, economics and environmental modelling wherever accumulated quantities, displacement or areas are required. This chapter also reinforces the broader calculus framework developed across Chapters 1 to 4, combining function analysis, algebra and interpretation.

Subchapters:

• 5A: Area under a curve
  
• 5B: Area between two functions
• 5C: Kinematics
• 5D: Problem solving by integration

By the end of this chapter, students will be able to use definite integrals to calculate areas, analyse motion and solve extended modelling problems with clear contextual interpretation.