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2021 HSC Mathematics Extension 1

Question 13(c):
Definite integral

3 marksdefinite integrals areas

The region enclosed by and is shaded in the diagram.

Find the exact value of the area of the shaded region.

The shaded region enclosed by y = 2 - |x| and y = 1 - 8/(4 + x squared).
The shaded region enclosed by y = 2 - |x| and y = 1 - 8/(4 + x squared).

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

  1. Both curves are even; the region is symmetric about the -axis. Intersection: for , .
  2. At : left ; right . So they meet at .
  3. Area .
  4. Primitive: .
  5. Evaluate to : .
  6. Total area .
  7. NESA: 3 marks full; 2 for a correct one-sided area involving the arctan curve; 1 for a correct integral setup, symmetry, or a relevant triangle area.

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 .

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Question 2

The curves and meet at and at where solves for . Write a symmetric integral for the enclosed area and give its antiderivative form.

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Question 3

Evaluate .

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