Level 3 Integration Walkthrough

2018 NCEA Level 3 Integration Question 1(e)

2018 Paper

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Question

The diagram shows the graph of

\[ f(x)=\frac12(e^x-1). \]
Area under an exponential curve to Q at k comma k The increasing curve f of x equals one half times e to the x minus one begins at the origin. The area under the curve from zero to k is shaded. A vertical line at x equals k reaches the point Q with coordinates k comma k. x f(x) k Q (k, k) f(x) = ½(eˣ − 1)

The point \(Q(k,k)\), where \(k>0\), lies on the curve. The shaded region is bounded by the curve, the \(x\)-axis, and the line \(x=k\).

Show that the shaded region has area \(\frac12k\).

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 area in terms of \(e^k\), then use the fact that \(Q(k,k)\) lies on the curve.

Step 1

Set up the area

Integrate the curve from the origin to the vertical boundary.

Show the first step’s working
\[ A=\int_0^k\frac12(e^x-1)\,dx. \]

Walkthrough overview

What this question practises

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

Method: Using a point on an exponential curve to prove an area result.

This is Question 1(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 using a point on an exponential curve to prove an area result. 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

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