Reduction formulas, arc length, surface area and improper integrals — the techniques for integrals that resist the standard methods.
Syllabus points covered
- Derive and use reduction formulas for the evaluation of definite integrals
- Use integration to find arc lengths and surface areas of revolution
- Evaluate improper integrals and recognise when they diverge
- Use hyperbolic and trigonometric substitutions
Common mistakes
- Deriving a reduction formula but not stating the base case needed to use it.
- Forgetting to change the limits after a substitution in a definite integral.
- Treating an improper integral as ordinary without taking the limit, and so missing a divergence.
- Losing the square root or the factor of 2π in arc length and surface area formulas.
Worksheets
Use these as soon as you've learned the topic and need to practise it. Start with the worksheet, check yourself against the answers, and only then look at the step-by-step solutions.
- Practice worksheetComing soon
- Worksheet answersComing soon
- Step-by-step solutionsComing soon
Topical past papers
Move on to these once you've worked through the worksheets and are ready for harder, exam-style questions. Real past-paper questions on this topic, with the official mark scheme and full worked solutions.
- Topical questions
- Mark scheme
- Step-by-step worked solutionsComing soon
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