I assume the region is being rotated around the y-axis (x=0).
Imagine a thin cylindrical shell of radius x and height h with the y-axis through its centre. The circumference of the cylinder (top and base) is 2πx.
If this cylinder is laid out as a thin rectangular lamina its dimensions would be 2πx by h, and its volume 2πxhdx, where dx is the thickness of the shell/lamina.
But h=e³ˣ-eˣ, so the volume is 2πx(e³ˣ-eˣ)dx. The sum of all such infinitesimally thin shells between x=0 and x=1 is:
2π∫₀¹x(e³ˣ-eˣ)dx.
Consider ∫xeⁿˣdx. Let u=x, then du=dx; and dv=eⁿˣdx, then v=eⁿˣ/n. Integrating by parts:
∫udv=uv-∫vdu=xeⁿˣ/n-(1/n)∫eⁿˣdx=xeⁿˣ/n-(1/n²)eⁿˣ.
So, setting n=1 and n=3:
2π∫₀¹x(e³ˣ-eˣ)dx=
2π[(xe³ˣ/3-(1/9)e³ˣ)+(xeˣ-eˣ)]₀¹=
2π(2e³/9+e-1/9+1)=2π(2e³/9+e+8/9)=50.71 cubic units approx.