First, remember that regardless of how crazy the graph of this function is, you are only interested in the part of it between x=0 and x=1.
If you are allowed, I would see what this looks like on a graphing calculator, just to get an idea of what you are working with. The part of this graph you are interested in is in quadrant I. In order to write your integrals, everything is given to you except you need to find where f(x) intercepts the y-axis, which you know you can find by solving for f(0). For this one, I would personally use the disk method for your rotation about the x axis and cylindrical shells about the y-axis.
I assume you have learned the techniques you need to solve these and you just needed an idea of where to start? If you need more help, just ask!