Variation 1 · answer guide

Definite integral — Absolute area not net

Question

Curves meet at and on , with on and on . The total shaded area between them is

This practice question was inspired by Question 2 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: find a suitable antiderivative, substitute the upper and lower bounds, and subtract in that order.Split at the interior crossing and orient each piece so the upper curve minus lower is used.
Simplify the previous line carefully, keeping exact values where possible.That yields .
Finish the calculation, then check that the result meets the question’s conditions.A single signed integral can cancel the two lobes.
Match the result to the choices and state the correct option.Therefore option B is correct.

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

Verification: If the two lobes have equal area, A gives while the true area is positive.

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