Integration

Integration Basics

  • Understand that integration is the reverse process of differentiation, also known as antiderivation.
  • Be familiar with the notation of integration: ∫ f(x) dx, where f(x) is the function to be integrated and dx is the differential element.
  • For definite integrals, understand the notation ∫ from a to b f(x) dx, where ‘a’ and ‘b’ are the limits of integration.
  • Grasp the power rule for integration, ∫ x^n dx = (1⁄(n+1))x^(n+1) + C, where n ≠ -1 and ‘C’ is the so-called ‘constant of integration’.

Calculating Indefinite and Definite Integrals

  • Be able to determine indefinite integrals of polynomials, exponential, and trigonometric functions.
  • Apply the power rule, as well as the rules for integrating e^x, sin(x), and cos(x).
  • Distinguish between indefinite and definite integrals: an indefinite integral is a function (or family of functions), while a definite integral is a number.
  • Calculate definite integrals using the Fundamental Theorem of Calculus: ∫ from a to b f(x) dx = F(b) - F(a), where F(x) is any antiderivative of f(x).

Rules and Techniques of Integration

  • Understand and utilise the basic rules of integration, including linearity of the integral and the power rule.
  • Practice integrating by substitution and by parts. Acknowledge these as useful techniques for integrating functions that don’t immediately appear suitable for direct application of the basic rules.
  • Recognise the necessity for different techniques of integration for different classes of functions, for example recognising when to use substitution, by parts, or trigonometric identities.

Applications of Integration

  • Apply integration to calculate areas under curves. Understand that ∫ from a to b f(x) dx calculates the net area between the curve y = f(x), the x-axis, and the vertical lines x = a and x = b.
  • Utilise integration to solve problems involving volumes of revolution. Recognise the disc method and cylindrical shell method as two common ways to do this.
  • Recall how the mean value theorem for integrals and the second fundamental theorem of calculus can be used to solve various problems, including calculation of averages and solving differential equations.