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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.

Integration Matcher

Stage: 5 Challenge Level: Challenge Level:1
This problem also appears on Underground Mathematics, where you can find more materials to support classroom use.

Why do this problem?

This problem offers students insights into differentiation, integration and the relationships between the two without needing to get involved with technical manipulations. It would be well suited to use as an introduction or summary to differentiation or integration. It is very good for giving intuitive meaning to the procedures and features of integration and differentiation, so would suit students with a range of technical skills.

The accurate charts can also be used as a problem involving fitting curves to equations. They can also be used to practise numerical integration.

Possible approach

You can print off the graphs here, or as accurate charts with scales here.

(One approach) Give the graphs to the class and say that they come in integral/differential pairs. For classes who have not yet met integration: "These graphs come in pairs. For each function $f(x)$ there is a graph of the function $A(x)$ where $A(x)$ is the area under the curve $y=f(t)$ between $t=0$ and $t=x$. Can you match each graph of a function $f$ with its corresponding graph of the area function $A$?"

Students could work in pairs for this activity. It is important that they give evidence for why they believe that each pair goes together. Students who have met the idea of integration as anti-differentiation could use this to check their pairings by looking for turning points mapping to a zero on the derivative graph. Students with a more sophisticated understanding should be expected to give the most rigorous answers.

(Another approach) Give the graphs out, but don't say that they come in integral differential pairs. Working in teams, ask how the graphs can be grouped. Ask the teams to share their ideas of groupings. Do any emerge as the best groupings? The notion of integrals/differentials is likely to emerge, but you might give a hint in this direction if groups are struggling. With this approach, be prepared to go with the flow of interesting pairing suggestions. Does the class as a whole agree that there is a 'best' overall grouping?

As always with matching questions, an emphasis should be placed on clear explanation of results. Guesses or poorly justified reasoning should be challenged most strongly!

Key questions

What are the key points on each graph?
Are there any statements you could make about the sign of the integral of each curve?
What evidence can you give for your statement? Does this convince you? Does it convince the rest of the class?
What algebraic forms might match these curves?

Possible extension

The extension in the question is challenging. Students might suggest the algebraic forms and give evidence to support their suggestion (after all -- the charts are only pictures. It is logically possible that the values come from any number of functions)

If the students were to invent a similar problem, which curves would they choose? Why?

Possible support

Encourage students to start with the 'easiest' curves. They could make sketches of what the area function would look like for each curve, and look to see if their sketch matches any of the cards.