Final answer:
The volume can be found using the disc method by integrating π(8 - 4 sec x)²dx over the given interval. This method applies the concept of summing the volumes of infinitely many thin discs formed by revolving the region around the x-axis.
Step-by-step explanation:
The volume generated by revolving a graphical region about the x-axis can be found using the disc method. The disc method involves slicing the solid into thin discs and summing the volumes of these discs. Each disc has a volume of πr²h, where r is the radius of the disc and h is the thickness. Since the solid described in the question is created by revolving the area between y = 4 sec x and y = 8 around the x-axis, the radius of each disc is the difference in the y-values (8 - 4 sec x).
The volume of a disc is therefore π(8 - 4 sec x)²dx. To find the total volume, we integrate this expression with respect to x over the given interval. This requires knowledge of integral calculus, specifically integration techniques for trigonometric functions.
As an example, let's consider the volume of a simple geometric shape such as a cylinder or a sphere, to illustrate these concepts in a straightforward case. For a cylinder, the volume is found by multiplying the area of the base circle (πr²) by the height (h). For a sphere, the volume is given by the formula (4/3)πr³.