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2023 HSC Mathematics Extension 1

Question 12(e):
Volume of revolution

4 marksvolumes of revolution

The region , bounded by the hyperbola , the line and the coordinate axes, is rotated about the -axis.

Find the volume of the solid of revolution formed. Leave your answer in exact form.

The supplied region R beneath the hyperbola, with rotation about the y-axis indicated.
The supplied region R beneath the hyperbola, with rotation about the y-axis indicated.

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

  1. Region : under from to , down to the axes. At , ; at , .
  2. Rotate about the -axis. Split: from the hyperbola for , plus cylinder of radius and height for .
  3. From : . Then .
  4. Expand: .
  7. using .
  8. So . Cylinder: .
  9. Total cubic units.
  10. NESA: 4 marks complete; 3 for hyperbola volume; 2 for correct integral setup; 1 for in terms of or two-volume decomposition.

Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.

The paper records 4 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

Region under from to , rotated about the -axis. Set up the -integral for the curved part.

See the step-by-step answer guide →

Question 2

Volume of cylinder radius , height , is

See the step-by-step answer guide →

Question 3

Simplify .

See the step-by-step answer guide →