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?

area between e to the x and the x axis is shaded and split in two between 0 and l n 2 and l n 2 and l n 3.
  • 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?

area between l n x and the y axis is shaded and split in two between 0 and l n 2 and l n 2 and l n 3