Find the volume of the solid obtained by rotating the region bounded by the curves $$ y = x^{2} $$ and $$ y = 4 $$ about the line $$ y = 5 $$ using the washer method.

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To find the volume of the solid obtained by rotating the region bounded by the curves $$ y = x^{2} $$ and $$ y = 4 $$ about the line $$ y = 5 $$ using the washer method, follow these steps:

1. **Determine the Points of Intersection:**
   The curves intersect where $$ x^2 = 4 $$, which gives $$ x = \pm 2 $$. So, the region of interest is between $$ x = -2 $$ and $$ x = 2 $$.

2. **Set Up the Washer Method:**
   The washer method involves integrating the area of washers (disks with holes) across the interval from $$ x = -2 $$ to $$ x = 2 $$.

- **Outer Radius ($$ R $$):** This is the distance from the line of rotation $$ y = 5 $$ to the curve $$ y = x^2 $$:
     $$
     R = 5 - x^2
     $$
   
   - **Inner Radius ($$ r $$):** This is the distance from the line of rotation $$ y = 5 $$ to the horizontal line $$ y = 4 $$:
     $$
     r = 5 - 4 = 1
     $$
   
   - **Volume Integral:**
     The volume $$ V $$ is given by:
     $$
     V = \pi \int_{-2}^{2} \left(R^2 - r^2\right) dx = \pi \int_{-2}^{2} \left((5 - x^2)^2 - 1^2\right) dx
     $$
     Expanding the integrand:
     $$
     (5 - x^2)^2 - 1 = 25 - 10x^2 + x^4 - 1 = x^4 - 10x^2 + 24
     $$
     So the integral becomes:
     $$
     V = \pi \int_{-2}^{2} \left(x^4 - 10x^2 + 24\right) dx
     $$
   
3. **Evaluate the Integral:**
   Since the integrand is even, we can simplify the computation:
   $$
   V = 2\pi \int_{0}^{2} \left(x^4 - 10x^2 + 24\right) dx
   $$
   Integrating term by term:
   $$
   \int_{0}^{2} x^4 dx = \frac{2^5}{5} = \frac{32}{5}, \quad \int_{0}^{2} -10x^2 dx = -10 \cdot \frac{2^3}{3} = -\frac{80}{3}, \quad \int_{0}^{2} 24 dx = 24 \cdot 2 = 48
   $$
   Combining these:
   $$
   V = 2\pi \left(\frac{32}{5} - \frac{80}{3} + 48\right) = 2\pi \left(\frac{96}{15} - \frac{400}{15} + \frac{720}{15}\right) = 2\pi \cdot \frac{416}{15} = \frac{832\pi}{15}
   $$

**Final Answer:**
The volume is 832⁄15 multiplied by π. In symbols,

Volume = (832 ⁄ 15) · π
The volume of the solid is $$ \frac{832}{15} \pi $$.

Find the volume of the solid obtained by rotating the region bounded by the curves and about the line using the washer method.

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