Problem 8. (10 points) Consider the region bounded by $y=0, x=0, x=1, y=2 e^{x}+1$.
- a) (5 pts) Find the volume of the solid obtained by rotating the region about the $x$-axis.
- b) (5 pts) Find the volume of the solid obtained by rotating the region about the $y$-axis.

Step 1 ：Given the region bounded by the equations $y=0$, $x=0$, $x=1$, $y=2 e^{x}+1$.

Step 2 ：We are asked to find the volume of the solid obtained by rotating this region about the x-axis and the y-axis.

Step 3 ：For part a), we can use the disk method to find the volume. The disk method formula is $V = \pi \int_{a}^{b} [f(x)]^2 dx$, where $f(x)$ is the function that describes the curve, and $a$ and $b$ are the limits of integration. In this case, $f(x) = 2e^x + 1$, $a = 0$, and $b = 1$.

Step 4 ：For part b), we need to use the shell method because the region is being rotated about the y-axis. The shell method formula is $V = 2\pi \int_{c}^{d} x f(x) dx$, where $f(x)$ is the function that describes the curve, and $c$ and $d$ are the limits of integration. However, we need to solve for $x$ in terms of $y$ to use this method. The equation $y = 2e^x + 1$ can be rearranged to $x = ln((y-1)/2)$. The limits of integration are the y-values where the curve intersects the y-axis, which are $y = 1$ and $y = 2e + 1$.

Step 5 ：By calculating the integrals, we find that the volume of the solid obtained by rotating the region about the x-axis is approximately 7.5814 and the volume of the solid obtained by rotating the region about the y-axis is approximately 5.5358.

Step 6 ：Final Answer: The volume of the solid obtained by rotating the region about the x-axis is approximately \(\boxed{7.5814}\) and the volume of the solid obtained by rotating the region about the y-axis is approximately \(\boxed{5.5358}\).