Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves y = x² and x = y² about the line y = 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 = x² and x = y² about the line y = 1, we can use the method of cylindrical shells.
The volume V is equal to the integral from a to b of 2πrh multiplied by the height difference dy, where r is the radius of the cylindrical shell and h is the height of the shell. In this case, the radius r is given by y - 1 and the height h is given by x² - y².
Therefore, the volume V is equal to the integral from 0 to 1 of 2π(y - 1)(x² - y²) dy. Solving this integral will give us the volume of the solid.