# How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by y=1/x, y=0, x=1, and x=2 about the y-axis?

Oct 11, 2016

Please see the explanation section, below.

#### Explanation:

Here is the region (blue) with a thin slice taken parallel to the axis of rotation. To set up for cylindrical shells. (The slice has black edges.)

The volume of a representative shell is $2 \pi r h \cdot \text{thickness}$

In this case, we have

radius $r = x$ (the dashed black line),

height $h = \frac{1}{x}$ and

$\text{thickness} = \mathrm{dx}$.

$x$ varies from $1$ to $2$, so the volume of the solid is:

${\int}_{1}^{2} 2 \pi x \left(\frac{1}{x}\right) \mathrm{dx} = 2 \pi {\int}_{1}^{2} \mathrm{dx} = 2 \pi {\left[x\right]}_{1}^{2} = 2 \pi$