Final answer:
The volume of the solid formed by rotating the region bounded by y = 1/x, x = 1, x = 8, and y = 0 about the x-axis is calculated using the disk method. The volume is found to be π * 7/8 cubic units.
Step-by-step explanation:
When considering the solid obtained by rotating the region bounded by y = 1/x, x = 1, x = 8, and y = 0 about the x-axis, we need to use the method of discs or washers to find the volume V of this solid. The general formula for the volume of a solid of revolution around the x-axis is:
V = π ∫ₓₑ f(x)^2 dx
For the given functions, we set up our integral from x=1 to x=8 with the cross-sectional area of the disk given by π(y^2). Since y = 1/x, the area is π(1/x)^2. Hence, the integral to find the volume V is:
V = π ∫₈₁ (1/x^2) dx
Integrating, we find:
V = π [ -1/x ]_1^8 = π [ (-1/8) - (-1/1) ] = π(1 - 1/8) = π * 7/8
Therefore, the volume of the solid is π * 7/8 cubic units.