## Problem

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 $1$?