Final answer:
To find the volume of the solid of revolution created by rotating the region bounded by x = (y - 3)^2 and x = 4 about y = 1, identify the bounds of integration, choose the cylindrical shells or disk/washer method and integrate over the bounds to get the volume.
Step-by-step explanation:
To find the volume of the solid formed by rotating the region bounded by the curves x = (y - 3)^2 and x = 4 about the line y = 1, we can use the method of cylindrical shells or the disk/washer method. The volume of a solid of revolution can be calculated using the formula V = π ∫ (outer radius)^2 - (inner radius)^2 dx for the disk/washer method and V = 2π ∫ (radius)(height) dy for the shell method.
First, we need to identify the bounds of integration by setting the two equations for x equal to each other and solving for y. Afterwards, select an appropriate method (such as cylindrical shells), and integrate the volume formula over the identified bounds. The radius in cylindrical shells will be y - 1, since we are rotating around y = 1, and the height will be the difference between the functions 4 and (y - 3)^2.
By carrying out the integration and simplifying, we obtain the volume of the solid of revolution.