Given curves:
[tex]64y=x^3, y=0, x=8[/tex]
The region is rotated about the line [tex]y=8[/tex]
Shell method formula : [tex]\int\limits^b_a {2\pi rh } \, dy[/tex]
Since the region is rotated about the line y= 8, we solve the given curve for x.
Take cube root on both sides.
The radius of the given region is [tex]y-8[/tex] and the height is [tex]x=4\sqrt[3]{y}[/tex].
The integral to find the volume is : [tex]\int_{8}^{16}2\pi (y-8)(4\sqrt[3]{y})dy[/tex]
