2021 HSC Mathematics Extension 1
Question 13(a):
Volume of revolution
A -metre-high sculpture is to be made out of concrete. The sculpture is formed by rotating the region between , and around the -axis.
Find the volume of concrete needed to make the sculpture.
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
- Outer curve: . Inner curve: . Height from to , but the inner solid only exists for .
- Volume outer solid of revolution minus inner hole: .
- Outer: .
- Inner: .
- Difference: .
- NESA: 3 marks full; 2 for evaluating one volume or a correct difference of -integrals; 1 for setting up a relevant volume integral or limits.
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
Same curves and , rotated about the -axis up to . Find the volume.
Question 2
Find the volume when from to is rotated about the -axis.
Question 3
Express the sculpture volume as a single washer integral in for the original problem, then evaluate.