Final answer:
The region is bounded by a cubic function, a vertical line, and the x-axis, forming a 2D area. This area is then revolved around the y-axis to find the volume of the solid, which is computed using the method of cylindrical shells via integration.
Step-by-step explanation:
The goal is to sketch the region bounded by the curves x=y³, x=8, and y=0, and then calculate the volume generated by revolving this bounded area around the y-axis. To begin, you would sketch the cubic function x=y³, which starts at the origin and curves upwards to the right. Since the area is also bounded by x=8, this vertical line will intersect the cubic curve at the point where y=∛8. The horizontal line y=0 (the x-axis) forms the bottom boundary of the region.
For calculating the volume of the solid formed by revolving this region around the y-axis, we use the method of cylindrical shells. The volume of a typical shell with radius y and height x (which in this case is y³) is given by 2πy · height · thickness. Therefore, the volume V of the entire solid is the integral from y=0 to y=∛8 of 2πy(y³) dy, which can be evaluated to find the total volume