Take a look at these diagrams. (There is a printable version of them available.)
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.
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 \(1\)?
