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Area L

By sketching a graph of a continuous increasing function, can you prove a useful result about integrals?

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Integral Equation

Solve this integral equation.

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Integral Sandwich

Generalise this inequality involving integrals.

Curved Square

Stage: 5 Challenge Level: Challenge Level:1

Why do this problem?

This problem draws together coordinate geometry, equations of circles, and surds, and can also be approached using integration.

Possible approach

Show the diagram (also available as a PowerPoint slide).
Give students time to study the diagram, and make notes about what they know and what they can work out. Pose the problem of finding the shaded area, and after some thinking time bring the class together to discuss possible strategies.
Two possible strategies are outlined in the following worksheets:
Sectors Method
Integration Method
You could outline the general methods to the class and give them time to solve the problem for themselves, offering the worksheet as a prompt if they get stuck.

Allow time at the end of the lesson for students to compare the different approaches.

Key questions

What information will we need to find the area?
What symmetries are present in the diagram? Does the area split up in any obvious ways?
Which would be the easiest arcs to work with? Why?
How can integration be used to find the area?

Possible extension

Find the areas of the other parts of the diagram.
Set up a similar problem using parabolas instead of circles.

Possible support

The worksheets suggest a suitable coordinate grid, and offer prompts for students to follow.