Use the method of cylindrical shells to find the volume v generated by rotating the region bounded by the given curves about the specified axis. y = x3, y = 8, x = 0; about x = 9

200π cubic units.

Step-by-step explanation:

Use the general method of integrating the area of the surface  generated by an arbitrary cross section of the region  taken parallel to the axis of revolution.

Here the axis  x = 9 is parallel to the y-axis.

The height of  one cylindrical shell = 8 - x^3.

The radius = 9 - x.

                                               2

The volume generated =  2π∫   (8 - x^3) (9 - x) dx

                                               0

= 2π ∫ ( 72 - 8x - 9x^3 + x^4) dx

             2

=      2 π [    72x - 4x^2 - 9x^4/4 + x^5 / 4  ]

            0

= 2 π  ( 144 - 16  - 144/4 + 32/4)

= 2 π * 100

= 200π.