Level 3 Integration Walkthrough

2017 NCEA Level 3 Integration Question 1(e)

2017 Paper

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Question

The mean value of a function \(y=f(x)\) from \(x=a\) to \(x=b\) is given by

\[ \text{Mean value}=\frac{\int_a^b f(x)\,dx}{b-a}. \]

Find the mean value of \(y=\sin^2x\) between \(x=0\) and \(x=\pi\).

Part of the graph of \(y=\sin^2x\) is shown below.

Graph of sine squared from zero to pi The non-negative curve y equals sine squared x starts at zero, reaches its maximum value one at x equals pi over two, and returns to zero at x equals pi. x y π 1 y = sin²x

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

Use the double-angle identity for \(\sin^2x\), integrate over one arch, then divide by the interval length.

Step 1

Use a double-angle identity

Rewrite the squared sine so it can be integrated directly.

Show the first step’s working
\[ \sin^2x=\frac12-\frac{\cos(2x)}2. \]

Walkthrough overview

What this question practises

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

Method: Finding a mean value using a squared-trigonometric identity.

This is Question 1(e) 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 finding a mean value using a squared-trigonometric identity. 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

← Question 1(d) Question 2(a) →