2024 HSC Mathematics Extension 1
Question 11(g):
Definite integral
The region is bounded by the curves , and the line as shown in the diagram (accessible description: for , the line lies above ; region is the area between them up to ).
Find the area of the region .
How to recognise this question
Question type: definite integrals areas
- The command and notation point to definite integrals areas.
- The requested response is a worked solution.
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
- On , , so area .
- Primitive: .
- Evaluate: .
- NESA 11(g): 3 marks correct; 2 marks for the primitive of or for finding the two relevant areas; 1 mark for considering , the triangle area, or a difference of areas.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 3 total marks 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
Find the area between and from to .
Question 2
Find the area between and from to .
Question 3
Find the area between and from to the first positive intersection is not required; instead evaluate .