Level 3 Integration Walkthrough

2018 NCEA Level 3 Integration Question 2(b)

2018 Paper

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Question

Find the value of \(k\), given that

\[ \int_1^k\sqrt{x}\,dx=\frac{52}{3}. \]

The integration must be shown.

First walkthrough idea

Focus to try first

Integrate \(\sqrt{x}\), evaluate at both limits, then solve the resulting power equation.

Step 1

Write a power and integrate

Use \(\sqrt{x}=x^{1/2}\) before applying the power rule.

Show the first step’s working
\[ \int_1^k x^{1/2}\,dx =\left[\frac23x^{3/2}\right]_1^k. \]

Walkthrough overview

What this question practises

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

Method: Finding an upper limit from a radical integral.

This is Question 2(b) 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 finding an upper limit from a radical integral. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

State or check the real-domain restriction and test for extraneous solutions after squaring.

Continue practising

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