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• NCEA on TKI
• NZQA website

Calculus 3.2 Integrate functions and use integrals to solve problems

Achievement criteria

The level of achievement that you reach is decided by the questions that you answer correctly.

Achievement

• You integrate functions and solve problems by integration, differential equations, or numerical methods which will include a selection from the following types:
  • axⁿ, where , including n = –1
  • polynomials in expanded forms
  • 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, for example kinematics
  • differential equations of the forms or for the above functions or situations where the variables are easily separable
  • finding areas under curves of the 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.
  
  
• Calculators may be used but you must be able to demonstrate the skill of integration.
• Diagrams may be provided for area and volume problems.

Achievement with Merit

• You need to reach Achievement.
• You use advanced integration techniques to find integrals and solve problems.
• You find integrals and use integration to 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, for example
    4x³ sin x⁴ or 3x⁵ + cosx
  • volumes of solids of revolution formed by rotating, around the x or y axis, the functions of the types listed for achievement
  • rates of change problems including kinematics
  • differential equations where students may be required to write their own differential equation to model a situation (applications could include growth and decay, inflation, Newton's Law of cooling and similar situations) for example y' = ky.

Achievement with Excellence

• You need to reach Achievement with Merit.
• You solve complex integration problems which may include finding:
  • areas between graphs of functions, other than polynomials, as listed above
  • volumes of solids of revolution formed by rotating around an axis parallel to the x or y axis
  • differential equations involving more difficult manipulation.

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