Final answer:
To find the volume of the solid created by rotating the region between x = y² and x = 1 about x = 1, one must use the method of cylindrical shells and evaluate the integral of 2π(1 - y²)y² from -1 to 1.
Step-by-step explanation:
To find the volume V of the solid obtained by rotating the region bounded by the curves x = y² and x = 1 about the line x = 1, we can use the method of cylindrical shells. The volume of a thin cylindrical shell with radius r, height h, and thickness Δr is given by the formula V = 2πrhΔr. The shells in this scenario have a variable radius r that is equal to 1 - y² (since we are rotating about x = 1) and a height of h that varies with y since x = y².
The volume of the solid is thus the integral of the volumes of these shells as y goes from -1 to 1 (the points where x = y² intersects x = 1). The integral is V = ∫_{-1}^{1} 2π(1 - y²)y² dy. Evaluating this integral will give us the volume of the rotated solid.