Final answer:
To find the volume generated when the region between the curves y = x³, y = 0, and y = 8 is rotated around the y-axis, use the method of cylindrical shells and integrate the volume of each cylindrical shell.
Step-by-step explanation:
To find the volume generated when the region between the curves y = x³, y = 0, and y = 8 is rotated around the y-axis, we need to use the method of cylindrical shells. We can integrate the volume of each cylindrical shell to find the total volume.
We want to find the volume between y = x³ and y = 8. The height of each cylindrical shell is given by the difference between the two curves, which is 8 - x³. The radius of each cylindrical shell is x, since we are rotating around the y-axis. The volume of each cylindrical shell is then given by V = 2πx(8 - x³)dx.
To find the total volume, we integrate V over the range of x values where y = x³ is below y = 8. This range can be found by setting x³ = 8 and solving for x, which gives x = 2. So, the integral becomes ∫(from 0 to 2) 2πx(8 - x³)dx.
After evaluating the integral, the total volume generated when the region between the curves is rotated around the y-axis is the answer to the integral.