There should be an upper limit on x because the x axis is an asymptote, making the surface area infinite. First we'll look at 0<x<1. The surface area can be seen as the sum of the surface areas of infinitesimally thin discs. The thickness of the disc is dx and the radius is e^-x. Therefore the surface area is 2(pi)e^-xdx. This is the surface area of the edge of the disc. Integrating this with the chosen limits we get -2(pi)[e^-x](0,1)=-2(pi)(e^-1-1)=2(pi)(1-1/e)=1.2642(pi)=3.9717 sq units.
Changing the upper limit to a we get 2(pi)(1-1/e^a).