Level 3 Integration Walkthrough

2017 NCEA Level 3 Integration Question 1(b)

2017 Paper

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Question

Use integration to find the area enclosed between the curve

\[ y=\frac{x^2+\sqrt{x}}{x} \]

and the lines \(y=0\), \(x=1\), and \(x=4\) (the area shaded in the diagram below).

Shaded area under y equals x plus one over square root x For positive x, the curve drops steeply from the vertical asymptote at x equals zero, reaches a minimum below y equals two, then rises. The region above the x-axis and below the curve from x equals one to x equals four is shaded. x y 1 4 y = x + x⁻½

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

Simplify the fraction into powers of \(x\), then integrate the curve from \(1\) to \(4\).

Step 1

Simplify the curve

Divide both numerator terms by \(x\), then write the radical as a power.

Show the first step’s working
\[ y=\frac{x^2}{x}+\frac{\sqrt{x}}{x} =x+x^{-1/2}. \]

Walkthrough overview

What this question practises

This 2017 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Using a definite integral to find an enclosed area.

This is Question 1(b) from the 2017 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 definite integral to find an enclosed area. 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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