Volume of revolution — Smaller constants

Question

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

This practice question was inspired by Question 12(e) in the 2023 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.; at , ; at , .
State the final answer clearly in the form the question requested.Curved volume plus cylinder if including .

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

Verification: Same decomposition as the 2023 solution.

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