Final answer:
The volume generated by rotating the region between y = 3x² and y = 18x - 6x² about the y-axis is found using the method of cylindrical shells, involving an integral that calculates the volume of each cylindrical shell formed by slices of the region.
Step-by-step explanation:
Method of Cylindrical Shells
The volume generated by rotating the region bounded by y = 3x² and y = 18x - 6x² about the y-axis can be found using the method of cylindrical shells. This method involves integrating the area of cylindrical shells created by slices of the region as they are revolved around the y-axis.
To set up the integral, we first need to find the points of intersection of the two curves by setting y = 3x² = 18x - 6x², which gives us the x-values. The integral is then evaluated from the first x-intercept to the second, representing each shell's volume as 2π times the average radius times the height (function value) times the thickness (dx).
Once the integral is evaluated, we have the total volume generated by the rotation. It is important to solve the integral correctly to ensure an accurate value for the volume.