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The washer method is a technique used in calculus to find the volume of a solid of revolution. This method involves imagining the solid being composed of stacked washers, or disks with holes in the center. By calculating the area of each washer and summing them using integration, we can determine the total volume.

The key to the washer method is determining the outer and inner radii of the washers:

• The outer radius (\(R_o\)) is the distance from the axis of rotation to the outer edge of the region.
• The inner radius (\(R_i\)) is the distance from the axis of rotation to the inner edge of the region.

In the exercise, the outer radius is defined by the curve \(y=\ln x\), and the inner radius is simply \(x=0\), leading to \(R_o = e^y\) and \(R_i = 0\).

The volume of a solid formed by revolving a region around an axis is an interesting problem often encountered in calculus. When the region is revolved, it creates a three-dimensional object, and its volume can be calculated using integration.

For solids of revolution, the volume (\(V\)) is calculated using the integral formula:\[V = \pi \int_{a}^{b}{\left(R_o^2-R_i^2\right) dy}\]This formula accounts for the difference in area between the outer circle and the inner circle (the hole in the washer), multiplied by the infinitesimal thickness (\(dy\)) of each washer. In the example, substituting our values of \(R_o\) and \(R_i\) gives:\[V = \pi \int_{0}^{2}{\left(\left(e^y\right)^2 - 0^2 \right) dy}\]

Calculus integration is a fundamental tool used to find various properties such as area, volume, and more, in mathematics. In the context of finding the volume of a solid of revolution, integration sums up the infinite number of infinitesimally small elements, like washers in this example.

The integral we set up in this exercise was:\[\pi \int_{0}^{2}{\left(e^{2y}\right) dy}\]This expression is evaluated by finding an antiderivative of \(e^{2y}\), which is \(\frac{1}{2} e^{2y}\). We calculate this over the interval from \(y=0\) to \(y=2\), generating the final result:\[V = \frac{\pi}{2}\left( e^{4}-1 \right)\]

Revolving a region around an axis is a way to create a symmetrical three-dimensional object. The process involves taking a two-dimensional region and "spinning" it around a line (axis of revolution) to form a solid.

In this problem, the region is revolved around the \(y\)-axis, defined by the boundaries \(y=0\), \(y=2\), \(y=\ln x\), and \(x=0\). When revolved around the \(y\)-axis, the small horizontal slices of the region become disks or washers, which make up the entire volume of the solid. This transformation from a flat region to a volume helps us visualize the nested layers that form the finished solid, and is crucial in applying the washer method.