Calculate the volume of the solid formed by rotating the area between $$ y = \sqrt{x} $$ and $$ y = 0 $$ from $$ x = 0 $$ to $$ x = 1 $$ about the line $$ y = -1 $$ utilizing the washer method.

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To calculate the volume of the solid formed by rotating the area between $$ y = \sqrt{x} $$ and $$ y = 0 $$ from $$ x = 0 $$ to $$ x = 1 $$ about the line $$ y = -1 $$ using the **washer method**, follow these steps:

### 1. Understanding the Washer Method

The washer method involves integrating the area of washers (disks with holes) along the axis of rotation. The volume $$ V $$ is given by:

$$
V = \pi \int_{a}^{b} \left(R^2 - r^2\right) dx
$$

where:
- $$ R $$ is the outer radius (distance from the axis of rotation to the outer curve),
- $$ r $$ is the inner radius (distance from the axis of rotation to the inner curve),
- $$ [a, b] $$ is the interval of integration.

### 2. Determine the Radii

- **Outer Radius ($$ R $$):** This is the distance from the axis of rotation $$ y = -1 $$ to the curve $$ y = \sqrt{x} $$.

$$
  R = \sqrt{x} - (-1) = \sqrt{x} + 1
  $$

- **Inner Radius ($$ r $$):** This is the distance from the axis of rotation $$ y = -1 $$ to the curve $$ y = 0 $$ (the x-axis).

$$
  r = 0 - (-1) = 1
  $$

### 3. Set Up the Integral

Substitute $$ R $$ and $$ r $$ into the washer formula:

$$
V = \pi \int_{0}^{1} \left[(\sqrt{x} + 1)^2 - (1)^2\right] dx
$$

Expand the integrand:

$$
(\sqrt{x} + 1)^2 - 1 = x + 2\sqrt{x} + 1 - 1 = x + 2\sqrt{x}
$$

So, the integral becomes:

$$
V = \pi \int_{0}^{1} \left(x + 2\sqrt{x}\right) dx
$$

### 4. Evaluate the Integral

Compute the integral term by term:

$$
\int x \, dx = \frac{1}{2}x^2 \quad \text{and} \quad \int 2\sqrt{x} \, dx = 2 \cdot \frac{2}{3}x^{3/2} = \frac{4}{3}x^{3/2}
$$

Thus:

$$
V = \pi \left[ \frac{1}{2}x^2 + \frac{4}{3}x^{3/2} \right]_{0}^{1}
$$

Evaluate from 0 to 1:

$$
V = \pi \left( \frac{1}{2}(1)^2 + \frac{4}{3}(1)^{3/2} \right) - \pi \left( 0 + 0 \right) = \pi \left( \frac{1}{2} + \frac{4}{3} \right) = \pi \left( \frac{3}{6} + \frac{8}{6} \right) = \pi \left( \frac{11}{6} \right)
$$

### 5. Final Answer

The volume of the solid is:

$$
V = \frac{11}{6} \pi
$$

**Answer:**  
After simplifying, the volume is (11⁄6) multiplied by π. Thus,

Volume = (11 ⁄ 6) · π
The volume of the solid is $$ \frac{11}{6} \pi $$.

Calculate the volume of the solid formed by rotating the area between and from to about the line utilizing the washer method.

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