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