What I did is
and I get , which I think is the right answer
but I can't seem to do it with the method suggested.
here you are trying to use the shell method. in this case you need to change the function from a function of x to a function of y, and change the limits as well. by the way, the lnx should not be squared when using this method, and you have to multiply by the radius instead. personally, i think this is more work than its worth. use the disk method