Here is the region we need to rotate:

11234-1xyOpen image in a new page

The graph of the area bounded by `y=x^3`, `x=0` and `y=4`.

And here is the volume generated when we rotate the region around the `y`-axis:

1-11234xyOpen image in a new page

The volume generated when revolving the curve bounded by `y=x^3`, `x=0` and `y=4` around the `y`-axis.

We first must express x in terms of y, so that we can apply the volume of solid of revolution formula.

If y = x3 then x = y1/3

The formula requires x2, and on squaring we have x2 = y2/3

`text[Vol] = pi int_c^d x^2 dy`

`=pi int_0^4 y^[2//3] dy`

`=pi [(3y^[5//3])/(5)]_0^4`



`=19.0\ text[units]^3`