See attachment for one such shell. The volume is given by the sum of infinitely many thin shells like this, each with radius [tex]x[/tex] and height determined by the vertical distance between the upper blue curve and the lower orange curve for any given [tex]x[/tex], i.e. [tex]8e^x-8e^{-x}[/tex].
The volume is then
[tex]2\pi\displaystyle\int_0^1x\left(8e^x-8e^{-x}\right)\,\mathrm dx=16\pi\int xe^x(e^{2x}-1)\,\mathrm dx=\dfrac{32\pi}9(e^3-4)\approx179.677[/tex]
