Level 3 Integration Walkthrough
2022 NCEA Level 3 Integration Question 1(d)
2022 Paper
Question
The graph below shows part of the function \(y=\frac{4}{\sqrt{3x-2}}\).
Find the value of \(k\) such that the shaded region has an area of \(8\).
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
Focus to try first
Because the curve stays above the axis, the shaded area is the definite integral from \(x=1\) to \(x=k\).
Step 1
Set up the area equation
The function is positive on the shaded interval, so ordinary area and signed area are the same here.
Show the first step’s working
Walkthrough overview
What this question practises
This 2022 walkthrough is part of AS91579 — Apply integration methods in solving problems.
Method: Setting up a shaded-area equation to solve for \(k\).
This is Question 1(d) from the 2022 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 a shaded-area equation to solve for \(k\). 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.