Can someone please explain to me where my error is in this problem because i should not be getting a negative volume.
Find the volume of the solid generating by revolving this region about the x-axis
$$y= 1 - |x|, y=0. $$
I have attached my work, but I just don't understand how I'm getting a negative answer
The integration is wrong.
For example, note that $(1+x)^2=1+\color{blue}{2x}+x^2$ and in general we do not have $(1+x)^2=1+x^2$.
Alternatively, to avoid expansion of the quadratic term, we can also do the following: $$\int_{-1}^0 (1+x)^2 \,dx= \left.\frac{(1+x)^3}{3}\right|_{x=-1}^{x=0}=\frac13$$
Also, by symmetry, the volume generated on the left of $x$-axis is equal to the volume generatd on the right.
Notice that one can also solve this problem (verify your solution) by using formula of volume of cone.