Level 3 Integration Walkthrough

2025 NCEA Level 3 Integration Question 3(e)

2025 Paper

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Question

Murray is planning to hang a piece of his art on a wall. This is shown in the diagram below.

f(x) = (x² + 3)² / 3 x y 1 2 3 Diagram is not to scale

The equation of the curved edge of the piece of art is \[ f(x)=\frac{1}{3}(x^2+3)^2. \]

Murray has researched a way to make the picture balance by using the following formula to find the \(x\)-value of the hanging position: \[ \frac{\int_0^3 xf(x)\,dx}{\int_0^3 f(x)\,dx}. \]

Use this formula to find the \(x\)-value of the hanging point.

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

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First walkthrough idea

Hint to try first

Treat the numerator and denominator as two separate definite integrals.

Step 1

Evaluate the numerator

With \(u=x^2+3\), the numerator becomes a straightforward power-rule integral.

Show the first step’s working
\[ \int_0^3 xf(x)\,dx=\int_0^3 \frac{x(x^2+3)^2}{3}\,dx \] \[ u=x^2+3,\qquad du=2x\,dx \]

With \(u=x^2+3\), the numerator becomes a straightforward power-rule integral.

Key result

\[ \left[\frac{(x^2+3)^3}{18}\right]_0^3=94.5 \]

Walkthrough overview

What this question practises

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

Method: Using weighted-average integrals to find a balance point.

This is Question 3(e) 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 weighted-average integrals to find a balance point. 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 3(d)