Level 3 Integration Walkthrough
2020 NCEA Level 3 Integration Question 3(b)
2020 Paper
Question
If \(\frac{dy}{dx}=\cos(2x)\), and \(y=1\) when \(x=\frac{\pi}{12}\), find the value of \(y\) when \(x=\frac{\pi}{4}\).
You must use calculus and give the results of any integration needed to solve this problem.
First walkthrough idea
Focus to try first
Integrate the gradient, use the initial condition to find \(C\), then evaluate at \(x=\frac\pi4\).
Step 1
Integrate the gradient
Reverse the chain rule for \(\cos(2x)\).
Show the first step’s working
Walkthrough overview
What this question practises
This 2020 walkthrough is part of AS91579 — Apply integration methods in solving problems.
Method: Integrating a gradient and using an initial condition.
This is Question 3(b) from the 2020 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 integrating a gradient and using an initial condition. Use the hints to plan the method, then repeat the question without hints and check each step.
Common mistake to avoid
Check the antiderivative by differentiating it, and handle constants and bounds explicitly.
Continue practising
- All 2020 Integration walkthroughs
- All AS91579 Integration years
- Practise more questions using this skill: Antidifferentiation