The region bounded by the curves \(y=x-1\), \(y=\ln (x)\), and \(y=1\) is revolved about the line \(y=1\).

To use the Washer Method to set up an integral or sum of integrals that would give the volume of the solid:

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

How many integrals will we need to express the volume of the solid using the Washer Method: \(\answer {2}\)

Express the volume of the solid using the Washer Method method:

\[ \int _{1}^{\answer {2}} \answer { \pi \left ( 1 - \ln (x) \right )^{2} - \pi \left ( 2-x \right )^{2} } \d x + \int _{\answer {2}}^{\answer {e}} \answer { \pi \left ( 1- \ln (x) \right )^{2} } \d x \]

To use the Shell Method to set up an integral or sum of integrals that would give the volume of the solid:

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

How many integrals will we need to express the volume of the solid using the Shell Method: \(\answer {1}\).

The integral that gives the volume of \(S\) is:
\[ \int _{\answer {0}}^{\answer {1}} \answer { 2\pi \left ( 1-y \right ) \left ( e^{y}-y-1 \right )} \d y \]
You should notice that the curves \(y=\ln (x)\) and \(y=x-1\) intersect when \(x=1\) and the corresponding \(y\)-value will be the lower limit of integration.