Further Calculus

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Further Calculus

Included in the detailed content statements on the material which must be included in A level specifications in further mathematics which makes up approximately 50% of the total content of A level further mathematics, we see the section on Further calculus.

Given that this is a compulsory topic, teachers of Further Maths can usefully look at specimen and practice materials for all the examination boards. We can also use technology to demonstrate and check answers. Additionally there are many useful resources offering notes and examples which are freely available.

WolframAlpha Volumes of Revolution
WolframAlpha can very easily be used to check any volumes of revolution and also gives a clear visualisation of the solid formed; note the option to show the surface or solid formed.

OCR Calculus Check In Test

OCR A Section Check In Test – Further Calculus

From OCR for syllabus A, the Check In Tests include Further Calculus (scroll down).

CIMT Reduction Formulae

The A Level notes from CIMT include some useful notes, activities and exercises for Further Mathematics, if we look at Chapter 8, Further Calculus in the Further Pure Mathematics section the we see that the chapter includes Volumes of revolution. We also have reduction formulae which is included in the optinal content for AQA, Edexcel and OCR Specification A.

For more notes, examples and exercises, not just for Calculus but many parts of the Further Maths specification, as well as CIMT, try AJ Hobson’s ‘Just the Maths’ or the Helm Project.

Just the Maths AJ Hobson

Many useful notes and examples can be found in AJ Hobson’s Just the Maths (individual pdfs hosted by UEA) or a complete pdf from the Math Centre: AJ Hobson’s ‘Just the Maths.

Helm Notes

Applications of Integration (14)

If you have not come across the HELM Project before, the project was designed to support the mathematical education of engineering students and includes an extensive collection of notes which include very clear worked examples. For easy access to these resources, the HELM Project Workbooks are hosted by Loughborough University’s Mathematics Learning Support Centre. Alternatively, the complete set is hosted by the Open University. To access the Open University resources you will need to create an account (easy and free), this will also give you access to the numerous free online courses.

For easy reference. a Notes and Examples page has been added to the Further Maths Series.

Improper Integrals University of Pennsylvania

Looking for resources on Improper Integrals, I came across this from the University of Pennsylvania, a clear presentation with worked examples.

Such integrals could be shown using WolframAlpha or Desmos. In Desmos, simply type b=infinity for the upper limit. For more on Integration with Desmos, see Learn Desmos: Integrals.
Improper Integral

Improper Integral

Edexcel have a Solution Bank and GeoGebra interactives for their various texts for Maths and Further Maths freely available online. If we look at Core Pure Mathematics Book 2 for example, you can see the various chapter links. These take you to the interactives for each chapter and to full worked solutions for each exercise. Looking at Chapter 4 for example on Volumes of Revolution, we can see two GeoGebra Interactives allowing exploration of volumes of revolution around the x and y axes.

Volumes of Revolution