Final answer:
To find the volume of the solid generated by revolving the region around the y-axis bound by y = 4x^2 and y = 4√x, use the shell method by integrating the shell volume formula from the intersections of the curves, x=0 to x=1. The shell volume is calculated as 2πx(height)(thickness), which translates to the cylinder volume concept of base perimeter times height times thickness.
Step-by-step explanation:
The question asks to find the volume of a solid generated by revolving a region around the y-axis, where the region is bounded by the curves y = 4x2 and y = 4√x. To solve this problem using the shell method, one should consider a representative shell at a distance x from the y-axis, with thickness dx, and height given by the difference in the y-values of the two curves at that x (4√x - 4x2). The volume of this shell is 2πx times the height of the shell (4√x - 4x2) times the thickness dx. Integrating this expression from x=0, where the two curves intersect, up to x=1, where they intersect again, gives the total volume.
The volume formula for a cylinder mentioned in a provided reference (V = Ah) helps in understanding that the volume of the shell is the product of the circumference (which is the 'perimeter of the base circle') and the height of the 'rectangular piece' when rolled out, multiplied by the thickness.
The given options (3/5)π, 3π, (12/5)π, (6/5)π seem to be possible results of an integration that would be performed to find the volume. The correct integration and evaluation will yield one of these values as the total volume.