Muluple Choice I poitit $$ x=0 $$ around the line $$ y=8 $$. $$ V=\pi \int_{0}^{1}\left[\left(8-x^{2}\right)-1^{2}\right] d x $$ $$ V=\pi \int_{0}^{3}\left[(8-x)^{2}-1^{2}\right] d x $$ $$ Y=\pi \int_{0}^{\pi}\left((8-t)^{2}-1^{2}\right) d x $$ none of these Previous

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Based on the information provided, it appears you're trying to determine the correct integral expression for calculating the volume $$ V $$ obtained by rotating a region around the line $$ y = 8 $$. Let's break down the potential scenarios and evaluate the given options.

### Understanding the Problem

Assuming you're rotating a region bounded by $$ y = x^2 $$, $$ x = 0 $$, and $$ x = 1 $$ around the line $$ y = 8 $$, the **washer method** is appropriate for this calculation.

### Washer Method Approach

1. **Identify the Outer and Inner Radii:**
   - **Outer Radius (R):** Distance from the line $$ y = 8 $$ to the top function $$ y = 1 $$, which is $$ R = 8 - 1 = 7 $$.
   - **Inner Radius (r):** Distance from the line $$ y = 8 $$ to the curve $$ y = x^2 $$, which is $$ r = 8 - x^2 $$.

2. **Set Up the Integral:**
   $$
   V = \pi \int_{0}^{1} \left[R^2 - r^2\right] dx = \pi \int_{0}^{1} \left[(7)^2 - (8 - x^2)^2\right] dx
   $$
   $$
   V = \pi \int_{0}^{1} \left[49 - (64 - 16x^2 + x^4)\right] dx = \pi \int_{0}^{1} \left[-15 + 16x^2 - x^4\right] dx
   $$

### Evaluating the Given Options

1. **Option 1:**
   $$
   V = \pi \int_{0}^{1} \left[(8 - x^2)^2 - 1^2\right] dx
   $$
   This doesn't correctly represent the outer and inner radii based on our washer method approach.

2. **Option 2:**
   $$
   V = \pi \int_{0}^{3} \left[(8 - x)^2 - 1^2\right] dx
   $$
   The limits of integration and the expressions within the integral do not match the problem's parameters.

3. **Option 3:**
   $$
   Y = \pi \int_{0}^{\pi} \left[(8 - t)^2 - 1^2\right] dx
   $$
   There's a mismatch in variables and limits, making this option incorrect.

4. **Option 4:**
   $$
   \text{None of these}
   $$
   Based on the above evaluations, none of the provided integral expressions correctly represent the volume using the washer method.

### Conclusion

Given the analysis, **none of the provided integral expressions accurately represent the volume of the rotated region** around the line $$ y = 8 $$.

**Answer:** none of these
none of these

Muluple Choice
I poitit around the line .

none of these
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Muluple Choice I poitit around the line . none of these Previous