Final answer:
The volume of the solid is obtained by integrating the difference in radii of the cylindrical shells formed by the rotation of the region about the line y = 4, from the curves y = 2e^{-x}, y = 2, up to x = 6.
Step-by-step explanation:
The student is tasked with finding the volume V of a solid generated by rotating a region bounded by specific curves and lines about another line. To solve this, we use the shell method or the disk method in calculus, which are techniques for finding volumes of solids of revolution. The process involves integrating the area of circular slices or cylindrical shells. Since y = 4 is the axis of rotation and is parallel to the x-axis, and since the region is between y = 2e−x and y = 2 up to x = 6, we evaluate the integral that represents the volume of the solid of rotation by taking into account the distance from the curves to the axis of rotation (y = 4). In this process, the integral of the difference between radii of the inner (2e−x rotated up to y = 4) and outer (y = 2 rotated up to y = 4) cylindrical shells multiplied by the height (2e−x - 2) from x = 0 to x = 6 gives the volume of the solid.