To find the volume of the solid obtained by rotating the region between the curves $y = x^2 + 1$ and $y = x + 3$ about the $x$-axis, we can use the method of cylindrical shells or the disk method. Here, we will use the disk method.

Steps:

1. Find the points of intersection:

Set the two functions equal to each other to find the points of intersection: $x^2 + 1 = x + 3$ $x^2 - x - 2 = 0$ $(x - 2)(x + 1) = 0$

Thus, the points of intersection are $x = 2$ and $x = -1$.

2. Set up the integral using the disk method:

The volume $V$ of the solid of revolution about the $x$-axis is given by: $V = \pi \int_{a}^{b} \left[ R(x)^2 - r(x)^2 \right] dx$ where $R(x)$ is the outer radius and $r(x)$ is the inner radius. For this problem, $R(x)$ is the upper function $y = x + 3$ and $r(x)$ is the lower function $y = x^2 + 1$.

3. Determine the limits of integration:

The limits of integration are from $x = -1$ to $x = 2$.

4. Set up and evaluate the integral:

$V = \pi \int_{-1}^{2} [(x + 3)^2 - (x^2 + 1)^2] \, dx$

Expanding the terms inside the integral:

$(x + 3)^2 = x^2 + 6x + 9$ $(x^2 + 1)^2 = x^4 + 2x^2 + 1$

Thus, the integral becomes:

$V = \pi \int_{-1}^{2} [x^2 + 6x + 9 - (x^4 + 2x^2 + 1)] \, dx$ $= \pi \int_{-1}^{2} [-x^4 - x^2 + 6x + 8] \, dx$

Now, we integrate each term separately:

$\int_{-1}^{2} -x^4 \, dx = \left[ -\frac{x^5}{5} \right]_{-1}^{2}$ $\int_{-1}^{2} -x^2 \, dx = \left[ -\frac{x^3}{3} \right]_{-1}^{2}$ $\int_{-1}^{2} 6x \, dx = \left[ 3x^2 \right]_{-1}^{2}$ $\int_{-1}^{2} 8 \, dx = \left[ 8x \right]_{-1}^{2}$

Evaluating each integral at the bounds:

$\left[ -\frac{x^5}{5} \right]_{-1}^{2} = \left( -\frac{2^5}{5} \right) - \left( -\frac{(-1)^5}{5} \right) = -\frac{32}{5} + \frac{1}{5} = -\frac{31}{5}$ $\left[ -\frac{x^3}{3} \right]_{-1}^{2} = \left( -\frac{2^3}{3} \right) - \left( -\frac{(-1)^3}{3} \right) = -\frac{8}{3} + \frac{1}{3} = -\frac{7}{3}$ $\left[ 3x^2 \right]_{-1}^{2} = 3(2^2) - 3((-1)^2) = 12 - 3 = 9$ $\left[ 8x \right]_{-1}^{2} = 8(2) - 8(-1) = 16 + 8 = 24$

Adding these results together:

$V = \pi \left( -\frac{31}{5} - \frac{7}{3} + 9 + 24 \right)$

Finding a common denominator (15):

$-\frac{31}{5} = -\frac{93}{15}$ $-\frac{7}{3} = -\frac{35}{15}$ $9 = \frac{135}{15}$ $24 = \frac{360}{15}$

So the integral evaluates to:

$V = \pi \left( -\frac{93}{15} - \frac{35}{15} + \frac{135}{15} + \frac{360}{15} \right)$ $= \pi \left( \frac{-93 - 35 + 135 + 360}{15} \right)$ $= \pi \left( \frac{367}{15} \right)$ $= \frac{367\pi}{15}$

Thus, the volume of the solid is:

$\boxed{\frac{367\pi}{15}}$

If you have any questions or need further details, let me know!

Here are 5 questions you might want to ask next:

1. How do you find the surface area of a solid of revolution?
2. What is the method of integration by parts in calculus?
3. Can you explain the Mean Value Theorem?
4. How do you solve a differential equation using separation of variables?
5. What are the properties of indefinite integrals?

Tip: When rotating regions around the axis, always visualize the solid and set up the correct limits of integration to ensure accurate results.