Here is the region we need to rotate:

And here is the volume generated when we rotate the region 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

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

=(3pi)/(5)[10.079-0]

=19.0\ text[units]^3

Get the Daily Math Tweet!
IntMath on Twitter