For Clarification, here is the Shell Method Formula that I was taught in school:
$V=2\pi\int_a^b r(x)h(x)dx$
$h(x)$ is defined as the $y$ coordinate of the function $f(x)$ at any value of $x$ between $x=a$ and $x=b$.
$a$ and $b$ are the lower and upper boundaries of the shape which will be rotated about the y-axis.
$r(x)$ is the distance between any $x$ (that is between $a$ and $b$) and the axis of revolution (the y-axis) which would make $r(x)=x$.
I created a graph on Desmos to visually depict what I mean by this. https://www.desmos.com/calculator/iazya3hpqn
Sometimes the axis of revolution is not the y-axis but rather $x=-3$ or $x=1$ in which case $r(x)=x+3$ or $r(x)=x-1$ given $a$ and $b$ are greater than $1$. This begs my question: Is there a case in which $r(x)$ would have to be nonlinear? Perhaps because the boundaries ($a$ and $b$) are not straight lines? I suppose the intended effect of changing $r(x)$ would be to match a nonlinear vertical boundary of the $3D$ object.