Q:

Working backwards from cylindrical shells

I have the volume of a solid, and I'm supposed to find the shape of the solid:
$$\int_0^1 2\pi(6-y)(1-y^2) \mathrm dy $$
This resembles a volume by cylindrical shells, but I'm not sure how to work backwards. It looks like 6-y is an adjusted rotation (i.e. not rotated around the x-axis). However, it's -y and not positive y, so I'm not sure how to find the original shape.

A:

The following little trick helps me.  Interchange the roles of $x$ and $y$. So our integral is now
$$\int_0^1 2\pi(6-x)(1-x^2)\,dx.$$
The change of letters brings us back to the standard setup where the shells method is first introduced, and where we have the most problem-solving practice.  We are looking at the curve $y=1-x^2$, a familiar downward-facing parabola.  Now it is essential to draw it. 
For $x$ going from $0$ to $1$, $1-x^2$ is the (vertical) distance from the point $(x,1-x^2)$ on the curve to the $x$-axis.  Draw a thin vertical strip of width "$dx$" down from $(x,1-x^2)$ to the $x$-axis.
And we are rotating about something that would be the $y$-axis if the $6-x$ was $x$.  But note that $6-x$ is the distance from the vertical line through $(x,f(x))$ to the vertical line $x=6$.  If we rotate our thin strip about the line $x=6$, we will get a shell of thirckness $dx$, radius $6-x$, and height $1-x^2$. "adding up" (integrating) will get us exactly the integral we wrote down.
Now go back to the original integral, by interchanging $x$ and $y$ again, which geometrically means reflecting across the line $y=x$.  Or now that we know what is going on, we can see that $x=1-y^2$ is a leftward-opening parabola, and that we are rotating the chunk to the right of the $y$-axis, from $y=0$ to $y=1$, about the line $y=6$. 

