Take a look at these diagrams. (There is a printable version of them available.)

Four diagrams which are described in the text below

Diagram 1 shows the graph of \(y=\dfrac1x\). \(I(a)\) is the shaded region under the graph between \(x=1\) and \(x=a\), where \(a>1\), so \(I(a)=\displaystyle{\int_1^a \dfrac{1}{x}\,dx}\).

Step 1: The shaded region in Diagram 4 has the same area as the shaded region in Diagram 1 (\(I(a)\)) after undergoing two stretches (either via Diagram 2 or via Diagram 3). Fill in the following information on each diagram where it is missing:

  • the scale factor that was used in the stretch;

  • the \(x\)-coordinate at each end of the shaded region;

  • the area of the shaded region in terms of \(I(a)\), and

  • the equation of the graph.

In the following applet, you can:

  • choose the value of \(a\)
  • stretch the graph (and the shaded region with it) in the \(x\)-direction by a scale factor of \(b\)
  • stretch the graph (and the shaded region with it) in the \(y\)-direction by a scale factor of \(c\)

The original area \(I(a)\) is shown, and the area of the stretched shaded region is also shown.

The original (unstretched) graph of \(y=\dfrac{1}{x}\) is shown as a dashed red line.

Step 2: Investigating the function \(I(x)\).

Recall from above that \(I(x)\) is the function that gives the area of the shaded region between \(1\) and \(x\) under the graph of \(y = \dfrac1x\), so \(I(x)=\displaystyle{\int_1^x \frac{1}{t}\,dt}\). (To avoid confusion, we have used \(t\) when writing this integral instead of \(x\), since \(x\) appears in the limits.)

  • Looking now at Diagram 4, can you identify the area \(I(b)\) on the diagram?

  • Are there any other \(I(x)\) values which you can easily identify on the diagram?

  • How are \(I(a)\) and \(I(b)\) related?

What do these answers suggest about the function \(I(x)\)? Have you seen anything with these properties before?

Some further questions

  • Using this applet or some graphing software, investigate values of \(I(x)\). Which value of \(a\) gives \(I(a)=1\)? What does this tell you?

  • What happens if one or both of \(a\) and \(b\) are less than \(1\)?