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.