Level 3 Integration Walkthrough

2021 NCEA Level 3 Integration Question 2(e)

2021 Integration

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Question

The diagram below shows the graph of a curve \(y=f(x)\), which satisfies the differential equation

\[ \frac{dy}{dx}=\frac{2}{ye^{0.5x}}. \]
x y 3 2 1 -1 -2 P Q

Points P and Q are the points on the graph of the curve that have \(x\)-coordinates of \(3\). What is the vertical distance between points P and Q?

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

Separate variables, use the two \(y\)-intercepts from the graph, then find the matching \(y\)-values when \(x=3\).

Step 1

Separate the variables

Rewrite \(e^{0.5x}\) in the numerator as \(e^{-0.5x}\).

Show the first step’s working
\[ \frac{dy}{dx}=\frac{2}{ye^{0.5x}} \] \[ y\,dy=2e^{-0.5x}\,dx \]

Walkthrough overview

What this question practises

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

Method: Solving a separable differential equation to find a vertical distance.

This is Question 2(e) 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 solving a separable differential equation to find a vertical distance. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.

Continue practising

← Question 2(d) Question 3(a) →