Level 3 Integration Walkthrough
2024 NCEA Level 3 Integration Question 3(d)
2024 Paper
Question
The graph below shows the function \(y=2\cos\left(\frac{x}{2}\right)\).
Find the value of \(k\) so that the shaded Area A will be equal to the shaded Area B.
You must use calculus and show the results of any integration needed to solve the problem.
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First walkthrough idea
Hint to try first
If Area A equals Area B, each one must be half of the total shaded area.
Step 1
Find the total area
\(\left[4\sin\left(\frac{x}{2}\right)\right]_0^{\pi}=4\).
Show the first step’s working
\(\left[4\sin\left(\frac{x}{2}\right)\right]_0^{\pi}=4\).
Key result
\[ 4 \]Walkthrough overview
What this question practises
This 2024 walkthrough is part of AS91579 — Apply integration methods in solving problems.
Method: Splitting a cosine area into two equal parts.
This is Question 3(d) 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.
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Learning summary
Review the method, not only the answer
This walkthrough helps you practise splitting a cosine area into two equal parts. 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.