Level 3 Integration Walkthrough
2022 NCEA Level 3 Integration Question 3(e)
2022 Paper
Question
The graph below shows part of the curve \(y=\frac{2}{x}\).
Points \(P\) and \(Q\) lie on the curve with \(x\)-coordinates \(k\) and \(3k\) respectively, where \(k>0\). Point \(R\) is such that \(PR\) is parallel to the \(x\)-axis and \(QR\) is parallel to the \(y\)-axis.
The shaded area can be written in the form \(a+b\ln c\), where \(a\), \(b\), and \(c\) are integers. Find the values of \(a\), \(b\), and \(c\).
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
Express the shaded region as a rectangle minus the area under the curve, then rewrite the logarithm in the requested \(a+b\ln c\) form.
Step 1
Find the rectangle area
The shaded region sits inside a rectangle whose width and height can both be read from the diagram.
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: Expressing a shaded region as rectangle minus logarithmic curve area.
This is Question 3(e) 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.
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Learning summary
Review the method, not only the answer
This walkthrough helps you practise expressing a shaded region as rectangle minus logarithmic curve 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.