To solve this problem using the shell method, we first need to understand the setup. The region is bounded by:

1. The line: $y = x + 12$
2. The parabola: $y = x^2$

The goal is to find the volume of the solid generated by revolving this region about the given lines. The formula for the volume using the shell method is:

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

Where:

• The radius is the distance from the axis of rotation to the shell.
• The height is the difference between the two functions $y = x + 12$ (the line) and $y = x^2$ (the parabola).

Let’s solve for each case:

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a. Revolving about the line $x = 4$

For this case, the radius of a shell at any point $x$ is the horizontal distance from the point to $x = 4$, which is $|4 - x| = 4 - x$ (since we are rotating to the right of $x = 4$).

The height of the shell is the difference between the line $y = x + 12$ and the parabola $y = x^2$, which is $(x + 12) - x^2$.

The volume integral is:

$V = 2\pi \int_{a}^{b} (4 - x) \times ((x + 12) - x^2) \, dx$

We need to find the bounds of integration, which correspond to the points where the line and parabola intersect. Set $x + 12 = x^2$:

$x^2 - x - 12 = 0$

Solving this quadratic equation:

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

Thus, $x = 4$ and $x = -3$ are the bounds.

The volume integral becomes:

$V = 2\pi \int_{-3}^{4} (4 - x) \times ((x + 12) - x^2) \, dx$

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b. Revolving about the line $x = -3$

For this case, the radius is the distance from the point $x$ to $x = -3$, which is $|x + 3| = x + 3$.

The height is still $(x + 12) - x^2$.

Thus, the volume integral is:

$V = 2\pi \int_{-3}^{4} (x + 3) \times ((x + 12) - x^2) \, dx$

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c. Revolving about the x-axis

Here, the axis of rotation is the x-axis. The radius is simply the value of $y$, which is the height from the x-axis to the curve. The height is the difference between the line and the parabola, $(x + 12) - x^2$.

Thus, the volume integral is:

$V = 2\pi \int_{-3}^{4} x \times ((x + 12) - x^2) \, dx$

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d. Revolving about the line $y = 16$

For this case, the radius is the vertical distance from the point $y = x^2$ or $y = x + 12$ to $y = 16$. The distance from $y = x^2$ to $y = 16$ is $|16 - x^2|$, and the distance from $y = x + 12$ to $y = 16$ is $|16 - (x + 12)| = 4 - x$.

The height is still $(x + 12) - x^2$, and the volume integral becomes:

$V = 2\pi \int_{-3}^{4} |16 - (x + 12)| \times ((x + 12) - x^2) \, dx$

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Summary of Steps:

• Find the bounds of integration by solving for the intersection points of the functions.
• Set up the volume integrals for each case using the formula for the shell method.
• Integrate over the appropriate bounds.

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Would you like to see the detailed integrations for each case, or do you have any specific part you want more clarification on?

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Here are 5 related questions you might find useful:

1. What is the concept of the shell method in calculating volumes?
2. How do you find the intersection points of two functions, such as $y = x + 12$ and $y = x^2$?
3. How does the radius of a shell change depending on the axis of rotation?
4. How do you set up the volume integral for rotation about an arbitrary line (not the x-axis)?
5. What are the general steps in solving volume integrals using the shell method?

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Tip: When dealing with the shell method, always carefully consider the distance from the axis of rotation and the difference in function heights. A solid understanding of these concepts is key for setting up correct integrals!