Level 3 Integration Walkthrough
2025 NCEA Level 3 Integration Question 3(d)
2025 Paper
Question
The graph below shows part of the curve
The area of the shaded region is \(\frac{9}{4}\) units\(^2\).
Prove that \(9p^3-2p^2-7=0\).
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
Rewrite the function as powers of \(x\) before integrating.
Step 1
Rewrite in integrable form
This makes the power rule straightforward to apply.
Show the first step’s working
This makes the power rule straightforward to apply.
Key result
\[ x^{-2}+6x^{-4} \]Walkthrough overview
What this question practises
This 2025 walkthrough is part of AS91579 — Apply integration methods in solving problems.
Method: Proving a cubic relation from a shaded area.
This is Question 3(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 proving a cubic relation from a shaded 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.