Level 3 Integration Walkthrough
2020 NCEA Level 3 Integration Question 2(e)
2020 Paper
Question
The graph shows the curve \(y=x+2\sqrt{x}-3\). The shaded area lies between the curve and the \(x\)-axis from \(x=0\) to \(x=4\), crossing the axis once within the interval.
Find the shaded area.
You must use calculus and give the results of any integration needed to solve this problem.
First walkthrough idea
Focus to try first
Find where the curve crosses the axis, then add the magnitudes of the two signed integrals.
Step 1
Find the x-intercept
Let \(u=\sqrt x\), solve the quadratic in \(u\), and retain \(u\ge0\).
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: Splitting geometric area where a curve crosses the axis.
This is Question 2(e) 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 splitting geometric area where a curve crosses the axis. 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.