Volume of revolution — A simple disk volume

Question

Rotate the region under , , about the x-axis.

This practice question was inspired by Question 12(b) in the 2025 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..
Simplify the previous line carefully, keeping exact values where possible..
Finish the calculation, then check that the result meets the question’s conditions..
State the final answer clearly in the form the question requested.Final answer:

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

Verification: The units are cubic and the integrand is non-negative.

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