To set up the integral for the volume of the solid formed when the region $R$ is revolved about the line $x = 4$ using the Shell Method, we proceed as follows:

Step 1: Analyze the given region $R$

The region $R$ is bounded by:

• $y = x^2$ (a parabola),
• $x = 0$ (the vertical line),
• $y = 4$ (a horizontal line).

This means that the region starts from $x = 0$ and extends to $x = 2$ (since $y = 4$ intersects $y = x^2$ at $x = 2$).

Step 2: Shell Method formula

The formula for the volume of a solid using the Shell Method when revolving around a vertical line $x = a$ is:

$V = 2\pi \int_{x_1}^{x_2} \text{(radius of shell)} \times \text{(height of shell)} \, dx$

In this case, we are revolving about $x = 4$.

• Radius of each shell: The distance from a point $x$ to the line $x = 4$. This is given by $|4 - x|$. Since the region is between $x = 0$ and $x = 2$, the radius simplifies to $4 - x$.

• Height of each shell: The height is the difference between the top and bottom of the region at a given $x$-value. The top of the region is given by $y = 4$, and the bottom of the region is given by $y = x^2$. So, the height of the shell is $4 - x^2$.

Step 3: Set up the integral

Now, applying the Shell Method formula:

$V = 2\pi \int_0^2 (4 - x) (4 - x^2) \, dx$

Final Answer:

$V = 2\pi \int_0^2 (4 - x) (4 - x^2) \, dx$

Would you like to solve this integral step by step? Or do you have any questions about the setup?

Related Questions:

1. How do you evaluate this integral?
2. How does the Disk Method compare to the Shell Method for this problem?
3. How do you apply the Shell Method when revolving around a horizontal line?
4. What would the volume be if the region was revolved around the