Level 3 Integration Walkthrough

2023 NCEA Level 3 Integration Question 3(c)

2023 Paper

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Question

The graph below shows the function \(y=5\sin(3x)\sin(x)\).

pi/3 x y y = 5 sin(3x) sin(x) Diagram is not to scale

Find the shaded area.

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 product-to-sum identity before integrating, then give the exact area and the decimal approximation.

Step 1

Apply the identity

This is the version that integrates cleanly.

Show the first step’s working

This is the version that integrates cleanly.

Key result

\[ \frac{5}{2}\left(\cos 2x-\cos 4x\right) \]

Walkthrough overview

What this question practises

This 2023 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Using product-to-sum in a shaded trigonometric area.

This is Question 3(c) from the 2023 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 product-to-sum in a shaded trigonometric 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 3(b) Question 3(d) →