I'm having trouble with finding the volume of y=x^2 from x=0, to x=1, revolving about the x axis. I'm having trouble visualizing it with this method. I can execute it with other methods, but not shell.
Using the washer method, it's easy to visualise them by comparing them to the rings of Saturn.
For the shells, pick a point on the y-axis at about 0.8
and draw a horizontal line until it touches the curve above the positive x-axis.
Rotate this line about the x-axis.
It traces out a cylinder.
You want the curved surface area of this cylinder (the surface area of the shell).
Now imagine doing the same for all the other horizontal lines from y=0 to y=1.
Integrate all the surface areas to get the volume (like layers of an onion).
All of the cylinders have to rest against the line x=1.
This is the case when we are calculating the VOR of the region
between the curve and horizontal axis using shells.
I've included 2 such cylinders, though there are "infinitely" many.
Therefore we need to subtract x from 1 to get the cylinder heights,
since the lines of length 1-x are being rotated around the horizontal axis.
Hope this helps.
Remember, these cylinders are "resting on their sides", not standing vertically.