2022 HSC Mathematics Extension 1
Question 13(b):
Volume of revolution
A solid of revolution is formed by rotating about the -axis the region bounded by the -axis and , where , between and .
Find the value of for which the volume is .
How to recognise this question
Question type: volumes of revolution
- The command and notation point to volumes of revolution.
- The requested response is a worked solution.
How to handle it: identify the radius and use the disk or washer formula with the correct variable and bounds.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Volume: .
- Use : .
- .
- At the upper limit , so .
- Set : (since ).
- NESA: 3 marks; 2 for primitive involving ; 1 for integral expression for the volume.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 3 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
For on , show the solid volume is .
Question 2
Using , find if .
Question 3
For , on , verify the volume is .