Variation 2 · answer guide

Definite integral — Three intersection splits

Question

On , changes sign at . Which expression always equals the total area between the graphs?

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.Total area is the integral of the absolute difference.
Finish the calculation, then check that the result meets the question’s conditions.Signed integrals over the whole interval cancel opposing regions.
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: Piecewise signed integrals with flipped lower regions are equivalent to .

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