Using the techique of chop or peel, setup an integral that represents the volume of the figure generated by the shaded area trapped by y=x^2+2, y=6 revolved around y=6


To set up the integral representing the volume of the solid formed by revolving the region bounded by \( y = x^2 + 2 \) and \( y = 6 \) around the line \( y = 6 \), we will use the **method of cylindrical shells** or **disk/washer method**.

### Step 1: Visualizing the Problem
The region in question is bounded:
- Below by the parabola \( y = x^2 + 2 \),
- Above by the horizontal line \( y = 6 \).

### Step 2: Choosing the Method
Since we are revolving around a horizontal axis (the line \( y = 6 \)), we can either use:
- The **washer method** (a vertical "peel" approach), which integrates with respect to \( x \).
  
In this case, let's proceed with the **washer method**.

### Step 3: Volume via Washer Method

The formula for the volume of revolution using the washer method is:

\[
V = \pi \int_a^b \left( R_{\text{outer}}^2 - R_{\text{inner}}^2 \right) dx
\]

Where:
- \( R_{\text{outer}} \) is the distance from the axis of revolution to the line \( y = 6 \),
- \( R_{\text{inner}} \) is the distance from the axis of revolution to the curve \( y = x^2 + 2 \).

#### Outer radius:
The distance from the axis \( y = 6 \) to the line \( y = 6 \) is zero, so the outer radius \( R_{\text{outer}} \) is simply 0. Since it's subtracted away in this setup, we instead focus on the inner curve.

#### Inner radius:
The distance from \( y = 6 \) to the curve \( y = x^2 + 2 \) is \( 6 - (x^2 + 2) = 6 - x^2 - 2 = 4 - x^2 \).

### Step 4: Limits of Integration
We need to find the limits of integration by determining the points where the curve \( y = x^2 + 2 \) intersects the line \( y = 6 \):
\[
x^2 + 2 = 6 \implies x^2 = 4 \implies x = -2 \text{ and } x = 2
\]
So the limits of integration are \( x = -2 \) to \( x = 2 \).

### Step 5: Setting up the Integral
The volume is given by the following integral:
\[
V = \pi \int_{-2}^{2} \left( (4 - x^2)^2 - 0 \right) dx
\]
Simplifying:
\[
V = \pi \int_{-2}^{2} (4 - x^2)^2 dx
\]

This represents the integral that gives the volume of the solid formed by revolving the region around \( y = 6 \).

---

Do you need further details or have any questions? Here are 5 related questions to further expand on this:

1. How do we evaluate the integral \( \int_{-2}^{2} (4 - x^2)^2 dx \)?
2. Can we use the cylindrical shells method instead of the washer method here?
3. How do we calculate the limits of integration when the axis of revolution is different?
4. How does the integral change if we revolve around the x-axis instead of \( y = 6 \)?
5. What are the differences between the disk and washer methods in general?

**Tip:** Always visualize the region and the axis of rotation clearly to choose the most convenient method for finding the volume.

Using the technique of chop or peel, setup an integral that represents the volume of the figure generated by the shaded area trapped by y=x^2+2, y=6 revolved around y=6

Learn how to set up an integral to find the volume of the solid formed by revolving the region bounded by y = x^2 + 2 and y = 6 around y = 6. This problem covers key concepts in integral calculus, including the washer method for solids of revolution. Suitable for Grades 11-12 and early college students.

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