Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves y = x³ and y = √x about the line y = x³, 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 = x³ and y = √x about the line y = x³, we can use the method of cylindrical shells. We need to find the limits of integration and the height of each shell. The limits of integration are from 0 to 1, which are the x-values where the curves intersect. The height of each shell is the difference between the two functions: (x³ - √x).
Using the formula for the volume of a cylindrical shell, V = 2πRHΔx, where R is the radius of the shell and Δx is the length of the shell, we can calculate the volume. The radius of each shell is y - x³, and the length of each shell is Δx = dx. Integrating the volume equation from 0 to 1 gives us the final volume.
V = ∫01 2π(x³ - √x)(dx)