2023 HSC Mathematics Extension 1
Question 4:
Definite integral
The diagram shows the graphs of the functions and .
It is known that , and .
What is the area between the curves and between and ?
How to recognise this question
Question type: definite integrals areas
- The command and notation point to definite integrals areas.
- The requested response is a multiple choice.
How to handle it: find a suitable antiderivative, substitute the upper and lower bounds, and subtract in that order.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- From the diagram, lies above on , so the geometric area equals .
- .
- Hence area .
- Using the signed integral of without care (e.g. or ) produces distractors; ignoring that area uses with on top also misleads.
- Answer: C. (NESA multiple-choice key: C.)
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 1 total mark for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
If , and on , the area between the curves is
Question 2
Given , , and on , the area between and is
Question 3
If but on , the area between the curves is