Question 717838
<pre>
That area in the first quadrant looks like this:

{{{drawing(600,400,-1.5,1.5,-.5,1.5, graph(600,400,-1.5,1.5,-.5,1.5,

(x^2-x^3)*(sqrt(x)/sqrt(x))(sqrt(1-x)/sqrt(1-x)) ),
rectangle(.54,0,.57,.15),locate(.55,.25,"(x,y)"),locate(.5,0,dx) 

)}}}

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

{{{drawing(600,400,-1.5,1.5,-.5,1.5, graph(600,400,-1.5,1.5,-.5,1.5,

(x^2-x^3)*(sqrt(x)/sqrt(x))(sqrt(1-x)/sqrt(1-x)) ),
rectangle(.54,0,.57,.15),locate(.55,.25,"(x,y)"),green(rectangle(-.54,0,-.57,.15)),
graph(600,400,-1.5,1.5,-.5,1.5,

(x^2+x^3)*(sqrt(-x)/sqrt(-x))(sqrt(1+x)/sqrt(1+x))*sqrt(sin(40x))/sqrt(sin(40x)) ),locate(.5,0,dx) 

)}}}

We will use the cylindrical shell method:

{{{drawing(600,400,-1.5,1.5,-.5,1.5, graph(600,400,-1.5,1.5,-.5,1.5,

(x^2-x^3)*(sqrt(x)/sqrt(x))(sqrt(1-x)/sqrt(1-x)) ),
locate(.55,.25,"(x,y)"),locate(.5,0,dx), 

graph(600,400,-1.5,1.5,-.5,1.5,

(x^2+x^3)*(sqrt(-x)/sqrt(-x))(sqrt(1+x)/sqrt(1+x))*sqrt(sin(40x))/sqrt(sin(40x)) ), rectangle(.54,0,.57,.15),green(rectangle(-.54,0,-.57,.15),
arc(0,.15,1.08,-.2-.1), arc(0,.15,1.14,-.23-.1) arc(0,0,1.14,-.23-.1,180,360), arc(0,0,1.08,-.2-.1,180,360)),blue(line(0,.15,.55,.15)), locate(.2,.24,r=x),
locate(.59,.12,h=y),locate(.5,0,dx) 

)}}}

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

Its circumference is 2&#960;r 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 = {{{int(2pi*r*h*dx,"",0,1)}}}{{{""=""}}}{{{2pi*int(r*h*dx,"",0,1)}}}{{{""=""}}}{{{2pi*int(x*y*dx,"",0,1)}}}{{{""=""}}}{{{2pi*int(x(x^2-x^3)*dx,"",0,1)}}}{{{""=""}}}{{{2pi*int((x^3-x^4)*dx,"",0,1)}}}{{{""=""}}}

{{{2pi*(
int(x^3*dx,"")

-
int(x^4*dx,"")

)

}}}{{{matrix(3,2,
"|",1,
"|","",
"|",0)}}}{{{""=""}}}{{{(2pi)(x^4/4-x^5/5)}}}{{{matrix(3,2,
"|",1,
"|","",
"|",0)}}}{{{""=""}}}{{{(2pi)(1^4/4-1^5/5)}}}{{{""-""}}}{{{(2pi)(0^4/4-0^5/5)}}}{{{""=""}}}{{{(2pi)(1/4-1/5)}}}{{{""=""}}}
{{{(2pi)(5/20-4/20)}}}{{{""=""}}}{{{(2pi)(1/20)}}}{{{""=""}}}{{{pi/10}}}   

Edwin</pre>