Find the volume *V* of the solid obtained by rotating the region bounded by the given curves about the specified line.

*y* = *x*3,   *y* = 1,   *x* = 2;

about *y* = −2
why is \pi \left(\left(\frac{\left(4\right)^7}{7}-\frac{1}{2}\left(4\right)^4-3\left(4\right)\right)-\left(\frac{\left(1\right)^7}{7}-\frac{1}{2}\left(1\right)^4-3\left(1\right)\right)\right) wrong

The expression you provided seems to be intended for calculating the volume of a solid of revolution, but it is incorrect for the given problem. Let's go step-by-step to clarify why and how the correct solution should be structured.

### Problem Breakdown
The curves that bound the region are:
- \( y = x^3 \) (cubic curve)
- \( y = 1 \) (horizontal line)
- \( x = 2 \) (vertical line)

You want to rotate this region about the line \( y = -2 \).

### Method: Washer Method

Since the rotation axis is horizontal, you should use the **washer method**. This method works by taking horizontal slices of the region, which are circular washers. The volume is found by integrating the difference of the squares of the outer and inner radii of these washers.

1. **Outer Radius (R(x))**: This is the distance from the line \( y = -2 \) to the curve \( y = 1 \) (the upper bound of the region):
   \[
   R(x) = 1 - (-2) = 3
   \]

2. **Inner Radius (r(x))**: This is the distance from the line \( y = -2 \) to the curve \( y = x^3 \) (the lower bound of the region):
   \[
   r(x) = x^3 - (-2) = x^3 + 2
   \]

3. **Washer Area**: The area of each washer is the difference between the outer circle's area and the inner circle's area:
   \[
   A(x) = \pi \left( R(x)^2 - r(x)^2 \right) = \pi \left( 3^2 - (x^3 + 2)^2 \right)
   \]

4. **Volume**: To find the volume, integrate the washer area with respect to \( x \) from 0 to 2 (the x-bounds):
   \[
   V = \int_0^2 \pi \left( 3^2 - (x^3 + 2)^2 \right) dx
   \]

### Why the Given Expression is Incorrect
Your expression:
\[
\pi \left(\left(\frac{4^7}{7}-\frac{1}{2}4^4-3(4)\right)-\left(\frac{1^7}{7}-\frac{1}{2}1^4-3(1)\right)\right)
\]
appears to be unrelated to the washer method and doesn't fit the geometry of the problem. Here are some reasons why it's incorrect:

- The powers of 7 and 4 do not arise naturally from the volume formula derived by the washer method for this problem.
- The numbers \( 4^7 \) and \( 1^7 \) suggest that you're using a polynomial-based approach, but the correct volume is based on the geometry of the problem, specifically circles (hence the square terms for radius calculations).
- The boundaries and setup of the problem do not involve such polynomial expressions.

### Correct Approach

The correct volume calculation is:

\[
V = \pi \int_0^2 \left( 9 - (x^3 + 2)^2 \right) dx
\]

We can now break down this integral for the exact volume calculation. Let me know if you'd like to proceed with this step or further clarification.

---

Do you want details or have any questions? Here are some related questions:

1. What is the washer method and when is it used in calculus?
2. How do you set up the limits of integration for a volume of revolution?
3. Why do we square the radius when calculating the volume of a solid of revolution?
4. How would the setup change if the axis of rotation was along a different line, such as \( y = 0 \)?
5. Can the disk method be used instead of the washer method in certain cases?

**Tip:** When calculating the volume of a solid of revolution, make sure to carefully analyze the distances from the axis of rotation to the curve to set up the correct radii for the washer formula.

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line: y = x^3, y = 1, x = 2; about y = -2. Why is π( (4^7/7 - (1/2)4^4 - 3(4)) - (1^7/7 - (1/2)1^4 - 3(1)) ) wrong?

Learn how to find the volume of a solid of revolution formed by rotating the region bounded by y = x^3, y = 1, and x = 2 about the line y = -2 using the washer method. This problem explores key calculus concepts, including volume integration and the washer method. Suitable for advanced high school or college-level students.

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