Investigation

Problem

In Curve match we asked you to match the curves on the diagram below with the following functions.

$y=\sqrt{x} \quad y=x^2 \quad y=x \quad y=\dfrac{x^2+x}{2} \quad \text{and } \quad y=2\sqrt{x}-x$

Now we’d like you to look at the different regions enclosed by the curves.

• Which do you think is the smallest region? How can you check?

• How big are the other regions?

• Try to explain why certain regions are the size they are.

Think about what techniques you can use and how you can reduce the number of calculations you need to do.

You might find it helpful to start by matching the curves, or you could check this in Curve match.