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.

Question

UpStudy AI · Accepted Answer

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.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Calculus Questions