Level 3 Integration Walkthrough

2017 NCEA Level 3 Integration Question 2(c)

2017 Paper

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Question

The diagram below shows the curve \(y=-x^2+3x+10\), and the line \(y=-x+14\), which is the tangent to the curve at the point \((2,12)\).

Shaded region bounded by a tangent, a parabola, and the x-axis The line y equals negative x plus fourteen is tangent to the downward parabola y equals negative x squared plus three x plus ten at two comma twelve. The shaded region lies below the tangent, above the parabola from x equals two to five, and above the x-axis from x equals five to fourteen. x y (2, 12) 5 14 y = −x + 14 y = −x² + 3x + 10

Calculate 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

Treat the shaded region as the area under the tangent from \(2\) to \(14\), minus the area under the parabola from \(2\) to \(5\).

Step 1

Split the shaded region

The source construction uses one large area under the tangent, then removes the area under the parabola.

Show the first step’s working

The shaded area is equal to the area under the tangent from \(x=2\) to \(x=14\), minus the area under the parabola from \(x=2\) to \(x=5\).

Walkthrough overview

What this question practises

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

Method: Finding the shaded area between a parabola and its tangent.

This is Question 2(c) 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 finding the shaded area between a parabola and its tangent. 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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