Level 3 Integration Walkthrough

2017 NCEA Level 3 Integration Question 3(c)

2017 Paper

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Question

Julia's friend Sarah believes that the equation of the curved border of the paved courtyard can be modelled by the function

\[ y=\frac{15x-15}{x+2}. \]
Courtyard area under a rational model The curve y equals fifteen x minus fifteen over x plus two begins at one comma zero and rises with decreasing slope. The courtyard area above the x-axis, below the curve, and between x equals one and x equals eleven is shaded. x y 1 11 y = (15x − 15)/(x + 2)

Use integration to find the area of the courtyard, shown in the diagram above.

You must use calculus and show the results of any integration needed to solve the problem.

The diagram is shown in its initial state. JavaScript adds any interactive controls and later walkthrough visuals.

First walkthrough idea

Focus to try first

Rewrite the rational function as a constant minus a reciprocal linear term, then integrate from \(x=1\) to \(x=11\).

Step 1

Rewrite the numerator

Create a multiple of \(x+2\), leaving a constant remainder.

Show the first step’s working
\[ 15x-15=15(x+2)-45, \] \[ y=\frac{15(x+2)-45}{x+2} =15-\frac{45}{x+2}. \]

Walkthrough overview

What this question practises

This 2017 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Rewriting a rational function to find the exact courtyard area.

This is Question 3(c) from the 2017 NCEA Level 3 Integration paper for AS91579 — Apply integration methods in solving problems. Use the guided hints to practise the method before revealing the full working.

Calc.nz is an independent learning resource. Compare questions, diagrams, and assessment information with the official NZQA resources for AS91579.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise rewriting a rational function to find the exact courtyard area. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check intersections, signs, and whether the question asks for signed area or total geometric area.

Continue practising

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