That area in the first quadrant looks like this:



and when it's rotated about the y-axis its cross section will look
like this:



We will use the cylindrical shell method:



The volume of a cylindrical shell is:
 
(its circumference) times (its height) times (its thickness).

Its circumference is 2πr and its thickness is dx.  Its radius, r, is the x
value of the point (x,y) which is just x. Its height, h, is just the y
value of the point(x,y) which is just y, (which we will replace by x²-x³):  

V = 


   

Edwin