Level 3 Integration Walkthrough
2021 NCEA Level 3 Integration Question 3(e)
2021 Integration
Question
The graph below shows the functions
where \(k\) is a constant greater than \(1\).
Show that the shaded area is \(\frac{k}{2}\left(k-1+\ln\frac{1}{k}\right)\).
You must use calculus and show the results of any integration needed to solve the problem. Clearly show each step of your working.
The diagram is shown in its initial state. JavaScript adds any interactive controls and later walkthrough visuals.
First walkthrough idea
Focus to try first
Find the intersection first. Then use top minus bottom from the intersection to the \(y\)-axis.
Step 1
Find the intersection
Set the two functions equal to get the left-hand limit of the shaded region.
Show the first step’s working
Walkthrough overview
What this question practises
This 2021 walkthrough is part of AS91579 — Apply integration methods in solving problems.
Method: Proving an exponential shaded-area formula in terms of \(k\).
This is Question 3(e) from the 2021 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 an exponential shaded-area formula in terms of \(k\). 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.