Let $y = 3^{x}+ 1, y = 0, x = 0$, and $x = 2$. Find the volume of the region revolved about each of the lines below.

1. $x$-axis

   Hint:

   Be sure to sketch the region before beginning.

2. $y$-axis

   Hint (b):

   If $y = 3^{x} + 1$, then $x=\log_3\left(y−1\right)$.

3. $y = −1$

   Hint (c):

   Notice, both the outer and the inner radii are $1$ unit longer than in part (a).

4. Explain why $\pi \int _ { a } ^ { b } ( R - r ) ^ { 2 } d x$ is the incorrect set-up for a washer given.

   Hint (d):

   True or False? $R^{2} − r^{2} = \left(R − r\right)^{2}$