• Integrate functions and use integrals to solve problems.

• Functions will include a selection from the following types:

  • axⁿ, where nR, including n = –1

  • polynomials in expanded form

  • exponential functions of the form

ae^{bx + c} (base e only)

• • trigonometric functions

  • rational functions such as .

• _{Problems will involve a selection from:}

  • rates of change problems, eg kinematics

  • differential equations of the forms y’ = f(x) or y” = f(x) for the above functions or situations where the variables are easily separable

  • finding areas under graphs of functions listed above

  • finding volumes of solids of revolution around the x axis using polynomial functions

  • finding areas using Simpson’s Rule or the Trapezium Rule.

• Diagrams may be provided for area and volume problems.

• Use advanced integration techniques to find integrals and solve problems.

• Integration will be based on a selection from:

  • products of trigonometric functions

  • simple algebraic substitutions

  • rational functions of the type

  • rational functions of the type .

• Problems will be selected from:

  • areas between graphs of polynomial functions

  • areas under graphs of functions listed for achievement with merit or combinations of those listed for achievement

eg 4x³ sinx⁴ or 3x⁵ + cosx

• • volumes of solids of revolution formed by rotating, around the x or y axis. The functions generated for integration will be of the types listed for achievement

  • rates of change problems including kinematics

  • differential equations where students may be required to write a differential equation to model a situation (applications could include growth and decay, inflation, Newton’s Law of Cooling and similar situations) eg.

• Solve more complex integration problem(s).

• Problem(s) may include finding:

  • areas between graphs of functions, other than polynomials, as listed above for achievement

  • volumes of solids of revolution formed by rotating around an axis parallel to the x or y axis

  • differential equations involving more difficult manipulation.