Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves y = sqrt(x) and y = 1 about the line y = sqrt(x) - 1, we can use the method of cylindrical shells.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves y = sqrt(x) and y = 1 about the line y = sqrt(x) - 1, we can use the method of cylindrical shells. We divide the region into thin vertical strips and rotate each strip around the given axis to create a cylindrical shell. The volume of each shell can be found using its height, circumference, and thickness.
By considering one such shell, we have its height as (1 - sqrt(x)), its circumference as 2π(sqrt(x) - 1), and its thickness as dx. Integrating the product of these quantities from x = 0 to x = 1 will give us the total volume of the solid.
Using the integral representation, the volume is given by:
V = ∫(0 to 1) 2π(sqrt(x) - 1)(1 - sqrt(x)) dx