Final answer:
The method of cylindrical shells is used to calculate the volume of a solid of revolution by integrating the volumes of concentric cylindrical shells. The volume of each shell is determined by the formula 2πrhδr. Integrating this over the interval that corresponds to the region's bounds will yield the solid's total volume.
Step-by-step explanation:
Method of Cylindrical Shells
To find the volume V generated by rotating the region bounded by the curves around the line 64y = x³, we apply the method of cylindrical shells. In this method, we imagine slicing the region into infinitesimally thin concentric cylinders (shells) and then summing up their volumes. The formula for the volume of a cylindrical shell with radius r and height h is Vshell = 2πrhδr, where δr represents the thickness of a shell. The process to obtain the total volume V is to integrate this expression over the interval of r corresponding to the bounds of the region.
To properly execute this technique, one must first express the radius and height of the cylindrical shells in terms of a single variable, typically x or y, based on the axis of rotation. In this scenario, because the rotation is around a vertical line, we'll likely express these in terms of y. Once the functions are set up, integration will yield the overall volume. Note that if the line of rotation was x = 64y, we would have a horizontal line, which requires us to adapt our expressions and limits of integration accordingly.
As an additional point of clarification, the provided text references unrelated contexts like the volume of a sphere and electric flux, which are not applicable to the cylindrical shell method for volume calculation. This exemplifies the importance of focusing solely on relevant information to solve the problem at hand.