Level 3 Integration Walkthrough

2017 NCEA Level 3 Integration Question 3(b)

2017 Paper

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Question

Julia wants to find an approximation of the area of a paved courtyard that she wishes to construct on her property. She takes some measurements and these are shown on the diagram below.

Measured paved courtyard for the trapezium rule The courtyard has a ten metre straight base split into five equal widths of two metres. Its measured heights at x equals zero, two, four, six, eight, and ten metres are zero, six, eight, ten, eleven, and twelve metres. A smooth curved border joins the tops of the measurements. 6 m8 m10 m11 m12 m2 m2 m2 m2 m2 m

Using these measurements, and the Trapezium rule, find an approximation of the area of paved courtyard.

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First walkthrough idea

Focus to try first

The five strips each have width \(2\) metres; count the two endpoint heights once and all four interior heights twice.

Step 1

Identify the measurements

The ordinates are \(0,6,8,10,11,12\) metres and the equal interval width is \(h=2\) metres.

Show the first step’s working
\[ h=2, \qquad y_0=0,\ y_1=6,\ y_2=8,\ y_3=10,\ y_4=11,\ y_5=12. \]

Walkthrough overview

What this question practises

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

Method: Applying the Trapezium Rule to courtyard measurements.

This is Question 3(b) 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 applying the Trapezium Rule to courtyard measurements. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.

Continue practising

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