Level 3 Integration Walkthrough
2019 NCEA Level 3 Integration Question 2(d)
2019 Paper
Question
The diagram shows the graph of \(y=\cos^2x\). The region under the curve between \(x=0\) and \(x=\pi\) is shaded.
Find the area of the shaded region.
You must use calculus and show the results of any integration needed to solve the problem.
First walkthrough idea
Focus to try first
Rewrite \(\cos^2x\) using the double-angle identity before integrating over \(0\le x\le\pi\).
Step 1
Use a trig identity
The identity \(\cos^2x=\frac12\cos(2x)+\frac12\) turns the square into integrable terms.
Show the first step’s working
Walkthrough overview
What this question practises
This 2019 walkthrough is part of AS91579 — Apply integration methods in solving problems.
Method: Using the \(\cos^2x\) identity to find shaded area.
This is Question 2(d) from the 2019 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 using the \(\cos^2x\) identity to find shaded 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.