Level 3 Integration Walkthrough
2024 NCEA Level 3 Integration Question 1(b)
2024 Paper
Question
The graph below shows the function \(y=40x(5x^2-3)^3\).
Find the shaded area.
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 shaded region runs from \(x=-0.4\) to \(x=0\), and the curve is above the \(x\)-axis on that interval.
Step 1
Set up the area
That matches the shaded region exactly.
Show the first step’s working
That matches the shaded region exactly.
Key result
\[ \int_{-0.4}^{0}40x(5x^2-3)^3\,dx \]Walkthrough overview
What this question practises
This 2024 walkthrough is part of AS91579 — Apply integration methods in solving problems.
Method: Setting up and evaluating a shaded polynomial area.
This is Question 1(b) 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 setting up and evaluating a shaded polynomial 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.