x=e^y², y=√ln(x). This curve meets the line y=1 when 1=√ln(x), that is when x=e.

The solid generated by revolution can be split into the volume of a unit cylinder (height and radius=1), which is π cubic units (x between 0 and 1), plus the difference between the volume of a cylinder with radius 1 and length e, which is πe cubic units, and π∫y²dx, for x between 1 and e.

So we have volume=π+πe-π∫y²dx[1,e].

Substitute ln(x) for y², we need to integrate ln(x)dx, so let dv=dx and u=ln(x), so v=x and du=dx/x. Integrating by parts we have uv-∫vdu=xln(x)-∫dx=xln(x)-x=x(ln(x)-1). When x=e this evaluates to 0, and when x=1 it evaluates to -1. So π∫y²dx=(0-(-1))π=π.

Therefore the volume is π+πe-π=πe cubic units.