Level 3 Integration Walkthrough

2018 NCEA Level 3 Integration Question 3(d)

2018 Paper

← Back to paper

Question

The diagram shows the graphs of \(y=x^2\) and \(y=\sqrt[3]{x}\).

Area between y equals x squared and y equals the cube root of x In the first quadrant, the cube-root curve lies above the parabola between their intersections at zero and one. The enclosed region between the curves is shaded. x y 1 1 y = x² y = ∛x

Find the area of the shaded region between the graphs.

You must use calculus and show the results of any integration needed to solve the problem.

The diagram is shown in its initial state. JavaScript adds any interactive controls and later walkthrough visuals.

First walkthrough idea

Focus to try first

Find both intersections, decide which curve is on top, then integrate top minus bottom.

Step 1

Find the intersections

Set the two function values equal.

Show the first step’s working
\[ x^2=x^{1/3} \quad\Longrightarrow\quad x^6=x, \] \[ x(x^5-1)=0. \]

Walkthrough overview

What this question practises

This 2018 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Intersections and area between a cube-root curve and a parabola.

This is Question 3(d) from the 2018 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.

Page updated .

Learning summary

Review the method, not only the answer

This walkthrough helps you practise intersections and area between a cube-root curve and a parabola. 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.

Continue practising

← Question 3(c) Question 3(e) →