Level 3 Integration Walkthrough

2018 NCEA Level 3 Integration Question 3(e)

2018 Paper

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Question

The diagram shows the graph of

\[ f(x)=(2x-1)^4. \]
Region bounded by a quartic curve, its tangent at Q, and the x-axis The quartic curve f of x equals two x minus one to the fourth touches the x-axis at P, one half comma zero, and passes through Q, one comma one. The tangent at Q crosses the x-axis at seven eighths. The region inside the curve, axis, and tangent is shaded. x y P (½, 0) ⅞ Q (1, 1) f(x) = (2x − 1)⁴ tangent

The curve meets the \(x\)-axis at \(P\), and the line shown is tangent to the curve at \(Q(1,1)\).

Find the area bounded by the curve, the \(x\)-axis, and the tangent at \(Q\).

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 the tangent at \(Q\), then subtract the area under that tangent from the area under the curve.

Step 1

Find the tangent gradient

Differentiate the quartic using the chain rule, then substitute \(x=1\).

Show the first step’s working
\[ f'(x)=8(2x-1)^3, \qquad f'(1)=8. \]

Walkthrough overview

What this question practises

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

Method: Subtracting the tangent area from the area under a quartic.

This is Question 3(e) 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.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise subtracting the tangent area from the area under a quartic. 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(d)