Level 3 Integration Walkthrough

2025 NCEA Level 3 Integration Question 1(d)

2025 Paper

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Question

The graph below shows part of the graph of the function \(y=4\sin(5x)\cos(3x)\).

π/6 x y y = 4 sin(5x) cos(3x) Diagram is not to scale

Find the shaded area under the curve between \(x=0\) and \(x=\frac{\pi}{6}\).

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

Use the product-to-sum identity \(2\sin A \cos B=\sin(A+B)+\sin(A-B)\).

Step 1

Rewrite the integrand

Multiply the identity by the outside factor \(2\) to get both sine terms with coefficient \(2\).

Show the first step’s working

Multiply the identity by the outside factor \(2\) to get both sine terms with coefficient \(2\).

Key result

\[ 2\sin(8x)+2\sin(2x) \]

Walkthrough overview

What this question practises

This 2025 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Using product-to-sum before integrating a shaded trig area.

This is Question 1(d) from the 2025 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 before integrating a shaded trig area. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check the antiderivative by differentiating it, and handle constants and bounds explicitly.

Continue practising

← Question 1(c) Question 1(e) →