IB DP Maths: AA HL复习笔记5.9.5 Modelling with Volumes of Revolution
Adding & Subtracting Volumes
When would volumes of revolution need to be added or subtracted?
How do I know whether to add or subtract volumes of revolution?
How do I solve problems involving adding or subtracting volumes of revolution?
- 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 3Use 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
Exam Tip
- 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
Worked Example
Modelling with Volumes of Revolution
What is meant by modelling volumes of revolution?
- 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
What modelling assumptions are there with volumes of revolution?
- 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
How do I solve modelling problems with volumes of revolution?
- 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)
- g. A question about a bucket may ask about its capacity
Exam Tip
- 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
Worked Example
Dimensions are in centimetres.
A mathematical model for a miniature vase is produced by rotating the region R through 2π radians around the x-axis.
Find the volume of the miniature vase, giving your answer in litres to three significant figures.
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