Final answer:
The question requires finding the volume of a solid by setting up integrals for rotation around various axes using shell and washer methods, based on the region bounded by y = 2x² and y = x³.
Step-by-step explanation:
The question asks to set up an integral to find the volume of the solid generated by rotating a region bounded by the curves y = 2x² and y = x³ around various axes using different methods such as shells and washers. First, we need to find the intersection points of the curves by setting 2x² = x³, which gives us the x-values where the region starts and ends. Depending on the axis of rotation and the method used, the integral set-up will vary. For example, when rotated about the x-axis using shells, we would set up an integral using the cylindrical shell method. However, when using washers, we need to calculate the volume of the annulus formed by the outer and inner radii at each x-value and integrate along the x-axis.
When rotating about the y-axis, a similar approach is taken but with respect to y-values using either the cylindrical shell method or the disk/washer method. Rotation about other lines such as x = -3 or y = -2 requires a translation of axes for setting up the correct integral. These problems typically involve calculus and geometric visualization skills to appropriately set up and calculate the desired volumes.