Final answer:
To find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = eˣ, and the line x = ln 8 about the line x = ln 8, we can use the method of cylindrical shells.
Step-by-step explanation:
To find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = eˣ, and the line x = ln 8 about the line x = ln 8, we can use the method of cylindrical shells.
First, we need to determine the limits of integration and the expression for the radius of each cylindrical shell. The limits of integration for x are from 0 to ln 8, since the region is bounded by the coordinate axes, y = eˣ, and x = ln 8. The radius of each cylindrical shell is given by the distance between the line x = ln 8 and the curve y = eˣ at a particular value of x. This distance can be calculated as R = |ln 8 - eˣ|.
Next, we need to determine the height of each cylindrical shell. The height can be calculated as h = eˣ - 0 = eˣ.
Finally, we can calculate the volume of each cylindrical shell using the formula for the volume of a cylinder: V = 2πrh. Integrating the volume of each cylindrical shell from x = 0 to x = ln 8 will give us the total volume of the solid.