Methods in calculus
Section overview:
The PowerPoint below provides an overview of material in this topic, including explanations and examples - for the full experience, open in a new tab then download it. All exercise references are to the new specification Pearson textbook - you can access the digital version for free or buy your own for extra practice.
Written summary notes are also given below for quick revision of key points, plus a selection of exam-style questions (with solutions) to test your understanding.
CP2-Chp3-MethodsInCalculus.pptxCP2-Chp4-VolumesOfRevolution.pptxNotes:
Section 1 Notes - Volumes of revolution (including parametrics)
Section 2 Notes - Improper integrals
Section 3 Notes - Calculus with inverse trig functions
Section 4 Notes - Further partial fractions, completing the square and trig substitutions
Section 5 Notes - Mean value of a function and summary of integrals
Summary questions:
Topic assessment 1 (with solutions) - Methods in calculus
Topic assessment 2 (with solutions) - Applications of integration
Extra resources:
Use the videos below to support your understanding of the topic. The ones on the left are specific to the course, building up to demos of exam-style questions; the ones on the right are for your own amusement, although the alternative outlook may provide you with a deeper overall understanding and appreciation of the topic.
In particular, 'The Essence of Calculus' video series is dotted throughout the relevant topic pages in A-level Maths and Further Maths, or you can find the full playlist here. These videos demonstrate a deeper cut of some of the most important ideas within A-level, so they are well worth a watch!
Topic videos:
- Improper integrals with infinite limits (convergent)
- Improper integrals with infinite limits (divergent)
- Improper integrals with the function undefined at a limit
- Improper integrals that cross a value where the function is undefined
- Differentiating arcsin(x)
- Differentiating arctan(x)
- Integrating using arcsin(x)
- Integrating using arctan(x)
- Partial fractions recap
- Partial fractions: denominator (ax+b)(cx2+d)
- Integrating: denominator (ax+b)(cx+d)
- Integrating: denominator (ax+b)(cx+d)2
- Integrating: denominator (ax+b)(cx2+d)
- Further integration using arcsin(x)
- Further integration using arctan(x)
- Substituting x=sin(u)
- Substituting x=tan(u)
- Substituting x=asin(u) or x=atan(u)
Extra fun! The Essence of Calculus - Epsilon-delta definitions and L'Hopital's rule (this is all really useful in higher level mathematics!):
The Essence of Calculus - A transformational view of calculus:
More fun with improper integrals:
Just some sweet, sweet integration...