Let \(R\) be the region in the \(xy\)-plane bounded by \(y=0\), \(y=\ln x\), \(y=2\), and \(x=0\). This exercise will walk you through setting up an integral using the Washer Method that will give the volume of the solid generated when \(R\) is revolved about the line \(x=-1\).

Since we are using the Washer Method, the slices must be:

parallel perpendicular
to the axis of rotation.

Slices that are perpendicular to the axis of rotation \(x=-1\) are:

vertical horizontal

Since the slices are horizontal, we must:

integrate with respect to \(x\). integrate with respect to \(y\).

Since we must integrate with respect to \(y\), we will use the result:

\[V = \int _{y=c}^{y=d} \pi \left (R^2-r^2\right ) \d y \]

to set up the volume. We must now find the limits of integration as express the outer radius \(R\) and the inner radius \(r\) in terms of the variable of integration \(y\).

The limits of integration are: \(c = \answer {0}\) and \(d = \answer {2}\).
For the curve given by \(y=\ln (x)\), we can solve for \(x\) to find \(x= \answer {e^y}\).

We thus have a helpful version of the picture of the region \(R\) below:

We see from the picture that both \(R\) and \(r\) are:

vertical distances horizontal distances
Since \(R\) is the distance from the axis of rotation to the outer curve, and this is a horizontal distance, we find \(R = x_{right}-x_{left}\).
\(x_{right} = e^y\) \(x_{right} = 1\) \(x_{right} = -1\)
\(x_{left} = e^y\) \(x_{left} = 1\) \(x_{left} = -1\)

So, \(R= \answer {e^y-(-1)}\).

Since \(r\) is the distance from the axis of rotation to the inner curve, and this is a horizontal distance, we find \(r = x_{right}-x_{left}\).

\(x_{right} = e^y\) \(x_{right} = 1\) \(x_{right} = -1\)
\(x_{left} = e^y\) \(x_{left} = 1\) \(x_{left} = -1\)

So, \(r= \answer {1-(-1)}\).

Using

\[V = \int _{y=c}^{y=d} \pi \left (R^2-r^2\right ) \d y, \]
we find that an integral that gives the volume of the solid of revolution is:
\[ V= \int _{y=\answer {0}}^{y=\answer {2}} \answer {\pi \left ((e^y-(-1))^2-(2)^2\right )}\d y \]