Final answer:
To find the volume of the solid generated by revolving the region bounded by y=e⁻²x, y=0, x=0, and x=ln8 about the x-axis, we can use the method of cylindrical shells. We integrate the volume of each infinitesimally thin cylindrical shell obtained by multiplying its circumference, height, and thickness.
Step-by-step explanation:
To find the volume of the solid generated by revolving the region bounded by y=e⁻²x, y=0, x=0, and x=ln8 about the x-axis, we can use the method of cylindrical shells. The volume of the solid can be calculated by integrating the circumference of each infinitesimally thin cylindrical shell multiplied by its height, which is the differential 'dx'. In this case, the height is given by the difference between the upper and lower curves (y=e⁻²x and y=0). The radius of each cylindrical shell is the x-coordinate.
An infinitesimally thin cylindrical shell has an infinitesimally small thickness of 'dx'. The volume of each cylindrical shell is given by dV = 2πx(e⁻²x-0)dx. Integrating this expression from x=0 to x=ln8 will give the desired volume of the solid.