For each x, the point $(x, \sqrt{x+1})$, rotated around the axis y= -1, moves in a circle with center at (x, -1) and radius $\sqrt{x+ 1}- (-1)= \sqrt{x+1}+ 1$. That circle has area $\pi r^2= \pi (\sqrt{x+1}+ 1)^2$. Thinking of that as a very thin disk with thickness "dx", its volume is $\pi (\sqrt{x+1}+ 1)^2 dx$. The total volume is the "sum" of all those disks: $\pi\int_{-1}^1(\sqrt{x+1}+1)^2 dx$.