Hello, ^_^Engineer_Adam^_^!
Your sketch may be incorrect . . .
Find the volume of the solid of revolution generated
when the given region is revolved about the indicated line.
Region bounded by: .y² = x³, .x = 0, .y = 8
. . revolved about x = 4.
Answer: .3456π/35 units³ Code:

8+     *
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:::* 
*+
 4
I prefer to use Cylindrical Shells.
. . V .= .2π ∫ (radius)(height) dx
The radius is: .4  x
The height is: .8  x^{3/2}
We have: . V . = . 2π ∫ (4  x)·(8  x^{3/2}) dx
. . . . . . . . . . .= . 2π ∫ (32  8x  4x^{3/2} + x^{5/2}) dx