The fourth option is correct.
See the attached image. The red cylinder represents a washer formed by the described revolution. Its volume is
[tex]\pi((\text{outer radius})^2-(\text{inner radius})^2)(\text{height})[/tex]
so when we integrate, we take
[tex]\displaystyle\pi\int_0^1((e^x+2)^2-3^2)\,\mathrm dx[/tex]
