Food for thought

## Warm-up ideas

Take a look at these graphs.

Say what you see…

Here are some things you might have noticed.

• There are two different graphs: each has a function and two $x$-coordinates given. There is an area shaded on each graph but they are in different places. One is between the curve and the $y$-axis and the other between the curve and the $x$-axis.

• Neither curve has any $y$-coordinates labelled so we can add these to the diagram. For $y = e^x$, the $y$-coordinates are $2$ and $3$. For $y = \ln x$ they are $\ln 2$ and $\ln 3$. So the same numbers appear in both graphs except the $x$ and $y$-coordinates have swapped over.

• The swapping of coordinates might remind us that there is a relationship between $e^x$ and $\ln x$. They are inverse functions of each other, which we can think about graphically as a reflection in $y = x$.

This symmetry around $y = x$ is not immediately obvious from the diagrams, as the scaling in the $x$ and $y$ directions is different on each sketch. It is important not to make assumptions about shape or symmetry based on sketch graphs.

• There are two rectangles on each graph with areas of $2 \times \ln 2$ and $3 \times \ln 3$ respectively. These areas are the same on both graphs as the $x$ and $y$ coordinates are reflected in $y=x$.

What can we say about the curved regions in the diagrams?

• The symmetry of the graphs implies that the two shaded areas are actually identical as they are reflected across $y=x$. They could be represented by the integral $\displaystyle{\int_2^{3} \ln x \, dx}$ or the integral $\displaystyle{\int_2^{3} \ln y \, dy}.$

Can you see which area is represented by which integral?

• We can also find the area of other regions bounded by the curves. For example, the area A can be represented by the integral

$\displaystyle{\int_{\ln 2}^{\ln 3} e^x \, dx}.$

As inverse functions have symmetry around $y = x$, we know that area A is the same as area B, shaded in the diagram below.

How would we represent B as an integral?