Integration and Area Worked Solution – 2022 NCEA Level 3 Q1(e) (AS91579)

Question

The graph below shows the functions \(y=(e^x)^2\) and \(y=3e^x+10\).

Find the exact value of the shaded area.

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

Find the intersection first, then integrate top minus bottom from the \(y\)-axis to that point.

Step 1

Find the intersection

You need the right-hand limit of the shaded region, so solve where the two curves meet.

Show the first step’s working

e^{2x}=3e^x+10 \] \[ \text{Let }u=e^x \] \[ u^2=3u+10 \] \[ u^2-3u-10=0 \] \[ (u-5)(u+2)=0 \] \[ u=5 \quad (\text{since }e^x>0) \] \[ e^x=5 \Rightarrow x=\ln 5

Walkthrough overview

What this question practises

This 2022 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Finding exact area between exponential curves.

This is Question 1(e) from the 2022 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.

Page updated 9 August 2026.

Learning summary

Review the method, not only the answer

This walkthrough helps you practise finding exact area between exponential curves. 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

← Question 1(d) Question 2(a) →