Level 3 Integration Walkthrough

2021 NCEA Level 3 Integration Question 1(b)(i)

2021 Integration

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Question

The gradient function of a curve is

\[ \frac{dy}{dx}=\frac{8}{x^3}. \]

Find the equation of the curve if it passes through the point \((1,3)\).

First walkthrough idea

Focus to try first

Integrate the gradient function, then use the point \((1,3)\) to find \(C\).

Step 1

Rewrite the gradient

The power rule is easier to apply when the denominator is written as a negative power.

Show the first step’s working
\[ \frac{dy}{dx}=\frac{8}{x^3}=8x^{-3} \]

Walkthrough overview

What this question practises

This 2021 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Integrating a gradient function and fitting the constant.

This is Question 1(b)(i) from the 2021 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 a gradient function and fitting the constant. 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(b)(ii) →