Let R be the region bounded by 
 , the x-axis, and the y-axis.

Write the integral used to determine the volume of the solid formed by revolving the region about the line y = 6 –x ^2.  (use disc/washer method - do not evaluate)

To determine the volume of the solid formed by revolving the region about the line \( y = 6 - x^2 \) using the washer method, we must first clarify the boundaries of the region. Since we have not been given the exact limits or function in the region directly in your problem statement, let's assume we are working with a typical region defined by a function \( f(x) \), bounded by the x-axis (where \( y = 0 \)) and the y-axis (where \( x = 0 \)).

The washer method involves calculating the difference between the outer radius (the distance from the axis of revolution to the further boundary of the region) and the inner radius (the distance from the axis of revolution to the closer boundary of the region).

### Outer Radius
The outer radius \( R_{\text{outer}}(x) \) is the distance from the line of revolution \( y = 6 - x^2 \) to the x-axis \( y = 0 \). This is simply the value of \( y = 6 - x^2 \), so the outer radius is:
\[
R_{\text{outer}}(x) = 6 - x^2
\]

### Inner Radius
The inner radius \( R_{\text{inner}}(x) \) is the distance from the axis of revolution \( y = 6 - x^2 \) to the function that defines the boundary of the region, \( y = f(x) \), if any is provided.

Now, let's write the general form of the volume integral:

### Volume Integral
\[
V = \pi \int_{a}^{b} \left[ R_{\text{outer}}(x)^2 - R_{\text{inner}}(x)^2 \right] \, dx
\]
Where:
- \( R_{\text{outer}}(x) \) is the distance from the axis of revolution to the outer boundary (in this case, \( 6 - x^2 \)),
- \( R_{\text{inner}}(x) \) is the distance from the axis of revolution to the inner boundary (if it exists),
- \( a \) and \( b \) are the bounds of integration, which would depend on the specific region.

Since the problem did not specify a particular function \( f(x) \) for the boundary, you may adjust \( R_{\text{inner}}(x) \) based on any additional details.

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Would you like further clarification on any part of this setup? Here are some related questions that could help expand your understanding:

1. How do we determine the bounds of integration \( a \) and \( b \)?
2. What happens if we rotate about a different line, like \( y = 0 \)?
3. Can you give an example where the function \( f(x) \) is explicitly provided for the region?
4. How does the washer method differ from the disk method when there's no inner radius?
5. How do we handle cases where the axis of revolution is horizontal or vertical?

**Tip:** When revolving regions using the washer method, always carefully check whether there's an inner radius to avoid mistakes in setup!

Let R be the region bounded by the x-axis, and the y-axis. Write the integral used to determine the volume of the solid formed by revolving the region about the line y = 6 –x ^2. (use disc/washer method - do not evaluate)

This page covers how to set up the integral to find the volume of a solid formed by revolving a region around the line y = 6 - x^2 using the washer method. Includes formulas for outer and inner radii and a detailed explanation of the washer method for undergraduate calculus students.

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