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The Washer Method is a powerful technique for finding the volume of a solid of revolution. This technique involves revolving a region around an axis and then calculating the volume by slicing the solid into thin disks or washers. Each washer is like a disk with a central hole, measured in terms of its inner and outer radii.

For the given problem, when the region bounded by the curves and lines is revolved around the x-axis, it results in a solid with washers as its cross-sections. The outer radius of each washer is defined by the line \( y = \sqrt{2} \), while the inner radius is given by the curve \( y = \sec x \). The differential volume of each washer is derived from the formula for the area of a circle, \( \pi (R^2 - r^2) \, dx \), where \( R \) and \( r \) are the outer and inner radii, respectively.

Thus, the Washer Method focuses on subtracting the volume of cylinders with radius \( r \) (the hole) from the volume of cylinders with radius \( R \), resulting in the total volume calculation of the solid.

The Volume Integral is the mathematical tool used to calculate the volumes of solids of revolution. In this exercise, using the Washer Method, we set up an integral that represents the volume of the solid created by revolving the bounded region about the x-axis.

The integral is constructed from the bounds of the interval of \( x \), which are \( -\pi/4 \) to \( \pi/4 \). The expression within the integral reflects the difference in the squared radii: \( (\sqrt{2})^2 - (\sec x)^2 \). This sets up the integral as:
\[V = \pi \int_{-\pi/4}^{\pi/4} \left( 2 - \sec^2 x \right) \, dx.\]

The integral will yield the volume of the solid when evaluated. This volume integral essentially sums up the volumes of the washers across the defined interval, providing the total volume of the revolved solid.

Integration Techniques play a crucial role in solving volume integrals accurately. Here, several techniques are applied to simplify and solve the integral for the total volume.

Firstly, the identity \( \sec^2 x = 1 + \tan^2 x \) is used to simplify the expression within the integral. This allows the integral to be split into two simpler parts:

• \( \pi \int_{-\pi/4}^{\pi/4} 1 \, dx \)
• \( - \pi \int_{-\pi/4}^{\pi/4} \tan^2 x \, dx \)

For the second part, another identity, \( \tan^2 x = \sec^2 x - 1 \), is used to further simplify the process. With these simplifications, each integral can be solved individually.

With basic calculus, evaluating the first integral yields \( \frac{\pi^2}{2} \), and for the second integral, the function \( \sec^2 x \) is integrated to give \( \tan x \). Finally, combining these results provides the complete volume of the solid, requiring careful evaluation to ensure accuracy.