Level 3 Integration Walkthrough

2023 NCEA Level 3 Integration Question 1(b)

2023 Paper

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Question

An object’s velocity can be modelled by

\[ v(t)=\sec^2 t, \]

where \(v\) is measured in km hr\(^{-1}\) and \(t\) is the time in hours from the start of timing.

Initially the object was \(3\) km from a point \(P\).

Find the distance of this object from the point \(P\) after \(\frac{\pi}{4}\) hours.

First walkthrough idea

Focus to try first

Velocity is the derivative of position, so integrate first, then use the starting position to find the constant.

Step 1

Find the position function

Position is found by integrating the velocity.

Show the first step’s working

Position is found by integrating the velocity.

Key result

\[ s(t)=\int \sec^2 t\,dt=\tan t + C \]

Walkthrough overview

What this question practises

This 2023 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Integrating velocity to get position, then using the starting distance.

This is Question 1(b) from the 2023 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 integrating velocity to get position, then using the starting distance. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check the antiderivative by differentiating it, and handle constants and bounds explicitly.

Continue practising

← Question 1(a) Question 1(c) →