Use horizontal rectangles and the complementary region

The region $R$ is bounded by the x-axis, y-axis, $y=1$ and $y=\ln x$, $x>0$. The diagram approximates its area by $A$, the total area of $n$ rectangles of height $1/n$, whose widths use the lower y-endpoint of each strip. Show that

$$A=\frac1n\left(\frac{1-e}{1-e^{1/n}}\right).$$

Another finite region $S$ is bounded by the x-axis, $x=e$ and $y=\ln x$, $x>0$. Explain how $A$ approximates the area of $S$, and whether this underestimates or overestimates it, giving a reason. Find the exact volume when $S$ rotates completely about the y-axis.