Level 3 Integration Walkthrough
2019 NCEA Level 3 Integration Question 1(d)
2019 Paper
Question
The graph shows the curve \(y=\frac{6}{\sqrt{3x+1}}\). Region A is the area under the curve from \(x=0\) to \(x=5\), and region B is the area under the curve from \(x=5\) to \(x=16\).
Show that the areas of regions A and B are equal.
You must use calculus and show the results of any integration needed to solve the problem.
First walkthrough idea
Focus to try first
Show the two areas are equal by evaluating \(\int_0^5 y\,dx\) and \(\int_5^{16}y\,dx\) with the same antiderivative.
Step 1
Write the function for integration
Convert the square-root denominator to a negative half-power.
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: Proving equal areas under a reciprocal-root curve.
This is Question 1(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 proving equal areas under a reciprocal-root curve. 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.