To find the volume of the solid generated by revolving the region bounded by the line $y = x + 12$ and the parabola $y = x^2$ about the x-axis using the shell method, we can proceed step by step:

Step 1: Set up the shell method formula

For the shell method, the formula for the volume is:

$V = 2\pi \int_{a}^{b} \left(\text{radius}\right) \times \left(\text{height}\right) \, dx$

• The radius is the distance from the axis of rotation (the x-axis) to the height of the shell, which is simply $y$ (since we revolve around the x-axis).
• The height is the difference between the upper curve $y = x + 12$ and the lower curve $y = x^2$, i.e., $(x + 12) - x^2$.

Step 2: Determine the bounds of integration

To find the limits of integration, we need to find the points where the line and the parabola intersect, which occur when:

$x + 12 = x^2$

Rearranging this equation:

$x^2 - x - 12 = 0$

Factoring the quadratic:

$(x - 4)(x + 3) = 0$

So, the points of intersection are $x = 4$ and $x = -3$. Therefore, the limits of integration are from $x = -3$ to $x = 4$.

Step 3: Apply the shell method

Now we can set up the integral for the volume:

$V = 2\pi \int_{-3}^{4} y \left[ (x + 12) - x^2 \right] \, dx$

Since the radius $y$ is simply $y = x$ and height is ( (x + 12) - x^2 X