Step 1 :Suppose that $R$ is the finite region bounded by $y=x$, $y=x+1$, $x=0$, and $x=3$.
Step 2 :We need to find the volume of the object we obtain when rotating $R$ about the $x$-axis.
Step 3 :The volume of the object obtained by rotating a region about the x-axis can be found using the formula for the volume of a solid of revolution, which is given by the integral from a to b of pi times the square of the function, dx.
Step 4 :In this case, the region is bounded by the lines $y=x$, $y=x+1$, $x=0$, and $x=3$. So, we need to find the integral from 0 to 3 of pi times the square of $(x+1)$ minus the square of $x$.
Step 5 :The volume of the object obtained by rotating the region about the x-axis is $12\pi$ cubic units.
Step 6 :\(\boxed{12\pi}\) is the exact value of the volume of the object we obtain when rotating $R$ about the $x$-axis.