Units 1 & 2 (Introduction to Integration)
Build foundational understanding of integration as the inverse of differentiation. Focus on computing basic definite and indefinite integrals of polynomial and related functions, interpreting integrals as area under curves, and applying integration to real-world contexts involving motion and accumulation.

Unit 2: Fundamentals of Integration

2.1 Understanding Integration

• Concept: integration as the reverse of differentiation

• Indefinite Integrals: ∫f(x) dx = F(x) + C

• Notation and Terminology: integrand, limits of integration, constant of integration (+C)

2.2 Basic Integration Rules

• Power Rule: ∫xⁿ dx = xⁿ⁺¹⁄(n+1) + C, for n ≠ –1

• Integration of Constants: ∫a dx = ax + C

• Linearity: ∫[af(x) + bg(x)] dx = a∫f(x) dx + b∫g(x) dx

2.3 Definite Integrals and Area

• Evaluating ∫ₐᵇ f(x) dx using antiderivatives

• Area Under a Curve: between the graph of f(x) and the x-axis

• Interpretation of Negative Area: understanding signed areas

• Units of Measurement: based on the context of f(x) (e.g. velocity → displacement)

2.4 Applications of Integration

• Motion Problems:

  • Velocity to displacement

  • Acceleration to velocity

• Accumulation Problems: total cost, volume, or quantity from rate functions

• Area Between Curves: ∫ₐᵇ [f(x) – g(x)] dx for vertical difference between two graphs

2.5 Use of Technology

• CAS Calculators:

  • Evaluating both definite and indefinite integrals

  • Checking manual integration

  • Graphing functions and visually interpreting area under curves