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.
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).
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.