Volume of revolution — Setup then evaluate

Question

Write an integral for the volume of on about the -axis, then evaluate it exactly.

This practice question was inspired by Question 12(b) in the 2024 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: identify the radius and use the disk or washer formula with the correct variable and bounds.Integral: .
Finish the calculation, then check that the result meets the question’s conditions.Value: .
State the final answer clearly in the form the question requested.Answer: .

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: after dividing by .

Any mark labels are a TestMum study aid, not an official NESA marking allocation.