Level 3 Integration Walkthrough

2018 NCEA Level 3 Integration Question 2(c)

2018 Paper

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Question

The diagram shows the graphs of \(y=\cos^2x\) and \(y=\sin^2x\).

First positive region between cosine squared and sine squared The solid cosine-squared curve starts at one and the dashed sine-squared curve starts at zero. The region between them from x equals zero to their first intersection at x equals k is shaded, with a dashed vertical guide at k. x y k y = cos²x y = sin²x

Find the first positive value of \(k\) shown by the diagram such that

\[ \int_0^k\left(\cos^2x-\sin^2x\right)\,dx=\frac12. \]

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

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First walkthrough idea

Focus to try first

Simplify the difference of the two squared trig functions before integrating.

Step 1

Use a double-angle identity

The difference of the two squared functions is a single cosine.

Show the first step’s working
\[ \cos^2x-\sin^2x=\cos(2x). \]

Walkthrough overview

What this question practises

This 2018 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: A double-angle identity and the first positive trig solution.

This is Question 2(c) from the 2018 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 a double-angle identity and the first positive trig solution. 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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