Suppose that $R$ is the finite region bounded by $y=x, y=x+1, x=0$, and $x=3$.
Find the exact value of the volume of the object we obtain when rotating $R$ about the $x$-axis.
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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.