Using the disk method, the volume is given by the integral
[tex]\displaystyle\pi\int_4^6\left(\frac{e^y}7\right)^2\,\mathrm dy[/tex]
See the attached image. It shows one such disk generated by the desired revolution. Its volume is [tex]\pi r^2h[/tex], where its radius is [tex]r=\frac{e^y}7[/tex] (the horizontal distance from the axis of revolution to the curve [tex]y=\ln7x[/tex]) and its height is an infinitesimally small change in [tex]y[/tex], included in the integral as the differential [tex]\mathrm dy[/tex].
We get
[tex]\displaystyle\pi\int_4^6\frac{e^{2y}}{49}\,\mathrm dy=\frac\pi{98}e^{2y}\bigg|_4^6=\frac{(e^{12}-e^8)\pi}{98}[/tex]
