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IB DP Maths: AA HL復習筆記5.9.5 Modelling with Volumes of Revolution

Category: IB課程, 教材筆記, 福利干貨 Date: 2022年7月15日上午11:24

Visualising the solid created becomes increasingly useful (but also trickier) for shapes generated by separate volumes of revolution
- Continue trying to sketch the functions and their solids of revolution to help

STEP 3

Use the appropriate volume of revolution formula for each part, evaluate the definite integral and add or subtract as necessary

The answer may be required in exact form

A sketch of the graph, limits, etc is always helpful, whether one has been given in the question or not
- Use your GDC where possible

5-9-5-ib-hl-aa-only-we1-soltn

Many everyday objects such as buckets, beakers, vases and lamp shades can be modelled as a?solid?of?revolution
The volume of revolution of the solid can then be calculated
An object that would usually stand?upright?can be?modelled?horizontally?so its?volume?of?revolution?can be found

The solids formed are usually the main shape of the body of the object
- For example, the handle on a bucket would not be included
The thickness of the solid is negligible relative to the size of the object
- thickness will depend on the purpose of the object and the material it is made from

Visualising and sketching the solid formed can help with starting problems
Familiarity with applying the volume of revolution fomulae
The volume of revolution may involve adding or subtracting partial volumes
Questions may ask related questions in context
- g. A question about a bucket may ask about its?capacity
  - this would be measured in litres
  - so a conversion of units may be required
  - (100 cm³?= 1 litre)

Remember to answer questions directly
- In modelling scenarios, interpretation is often needed after finding the 'final answer'
Modelling questions often ask about assumptions, criticisms and/or improvements
Examples
- it is assumed the thickness of the material an object is made from is negligible
- a 'smooth' curve may not be a good model if the item is being made from a rough material
- other things may significantly reduce the volume found and impact conclusions
  - e.g. ?Stones, plants and decorations placed in an aquarium will reduce the volume of water needed to fill it - and hence the number/size/type of fish it can accommodate may be impacted