Level 3 Integration Walkthrough

2024 NCEA Level 3 Integration Question 2(d)

2024 Paper

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Question

The graph below shows part of the graph of the function \(y=\sin^2x\).

y = 1 y = sin²x x = -π/2 x = π/2 x y Diagram is not to scale

Find the shaded area enclosed between the lines \(y=\sin^2x\), \(y=1\), \(x=-\frac{\pi}{2}\), and \(x=\frac{\pi}{2}\).

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

Hint to try first

The top function is \(y=1\), so the area starts as \(\int (1-\sin^2x)\,dx\).

Step 1

Set up the area

The line \(y=1\) sits above the curve \(y=\sin^2x\) on this interval.

Show the first step’s working

The line \(y=1\) sits above the curve \(y=\sin^2x\) on this interval.

Key result

\[ \int_{-\pi/2}^{\pi/2}(1-\sin^2x)\,dx \]

Walkthrough overview

What this question practises

This 2024 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Using trig identities in a shaded \(\sin^2x\) area.

This is Question 2(d) from the 2024 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 using trig identities in a shaded \(\sin^2x\) 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

← Question 2(c) Question 2(e) →