Question 356351
Let's look at a cone (in cross section). 
{{{drawing(300,300,-10,10,-10,10,circle(0,-8,0.3),circle(-5,8,0.3),circle(5,8,0.3),locate(2,9,r1),locate(1,1,r2),locate(0.2,-2.5,h1),locate(0.2,4.5,h2),
line(0,-8,5,8),line(5,8,-5,8),circle(2.5,0,0.3),circle(-2.5,0,0.3),line(-2.5,0,2.5,0),line(-5,8,0,-8), line(0,-8,0,8))}}}
Actually there are two cones in the picture.
The large cone with a radius of r2 and height of {{{h1+h2}}} and a smaller cone with a radius of {{{r1}}} and height {{{h1}}}. The figure in between would be the tapered bucket if you rotated the cross section about the axis.
So the volume of the tapered bucket would be the volume of the large cone minus the volume of the small cone.
The volume of a cone is,
{{{V[bc]=(pi/3)R^2*H}}}
Big Cone:
{{{V[sc]=(pi/3)r2^2*(h1+h2)}}}
Small Cone:
{{{V=(pi/3)r1^2*(h1)}}}
Tapered Bucket:
{{{V[tp]=(pi/3)r2^2*(h1+h2)-(pi/3)r1^2*(h1)}}}
There is also a relationship between the radii and the heights.
{{{(r1)/(h1)=(r2)/(h1+h2)}}}
{{{r1*h1+r1*h2=r2*h1}}}
{{{h1(r2-r1)=r1*h2}}}
{{{h1= (r1*h2)/(r2-r1) }}}
{{{h1+h2=(r1/(r2-r1))+h2}}}
{{{h1+h2=(r1/(r2-r1))h2+h2*((r2-r1)/(r2-r1))}}}
{{{h1+h2= (r2/(r2-r1))*h2}}}
Substituting,
{{{V[tp]=(pi/3)(r2^2*(h1+h2)-r1^2*(h1))}}}
{{{V[tp]=(pi/3)(r2^2*((r2/(r2-r1))*h2)-r1^2*((r1*h2)/(r2-r1)))}}}
{{{V[tp]=(pi/3)(h2/(r2-r1))(r2^3-r1^3))}}}
Dividing by {{{(r2-r1)}}}
{{{highlight(V[tp]=(pi/3)(h2)(r2^2+r1*r2+r1^2)))}}}