Calculate the volume of the solid generated by revolving the region bounded by the curve $$ y = x^2 $$ and the line $$ y = 4 $$ around the x-axis.

Question

UpStudy AI · Accepted Answer

To calculate the volume of the solid generated by revolving the region bounded by the curve $$ y = x^2 $$ and the line $$ y = 4 $$ around the x-axis, we'll use the **washer method**. Here's a step-by-step solution:

### 1. **Identify the Points of Intersection**
First, determine where the curves $$ y = x^2 $$ and $$ y = 4 $$ intersect:

$$
x^2 = 4 \implies x = \pm 2
$$

So, the region of interest is between $$ x = -2 $$ and $$ x = 2 $$.

### 2. **Set Up the Washer Method Integral**
When revolving around the x-axis, the volume $$ V $$ can be calculated using the washer method:

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

Where:
- $$ R(x) $$ is the outer radius (distance from the x-axis to the upper curve).
- $$ r(x) $$ is the inner radius (distance from the x-axis to the lower curve).

In this case:
- The **outer curve** is $$ y = 4 $$, so $$ R(x) = 4 $$.
- The **inner curve** is $$ y = x^2 $$, so $$ r(x) = x^2 $$.

### 3. **Express the Integral**
Substitute $$ R(x) $$ and $$ r(x) $$ into the integral:

$$
V = \pi \int_{-2}^{2} \left[ 4^2 - (x^2)^2 \right] dx = \pi \int_{-2}^{2} \left[ 16 - x^4 \right] dx
$$

Since the integrand is even, we can simplify the calculation by integrating from 0 to 2 and then doubling the result:

$$
V = 2\pi \int_{0}^{2} \left[ 16 - x^4 \right] dx
$$

### 4. **Compute the Integral**
Evaluate the integral:

$$
\int_{0}^{2} \left[ 16 - x^4 \right] dx = \left[ 16x - \frac{x^5}{5} \right]_0^2 = \left( 16 \times 2 - \frac{32}{5} \right) = 32 - \frac{32}{5} = \frac{160 - 32}{5} = \frac{128}{5}
$$

### 5. **Calculate the Volume**
Multiply by $$ 2\pi $$:

$$
V = 2\pi \times \frac{128}{5} = \frac{256}{5} \pi
$$

### **Final Answer**
The volume of the solid is:

$$
V = \frac{256}{5} \pi \text{ cubic units}
$$

Answer: Its volume is 256⁄5 multiplied by π, that is V = (256⁄5) · π cubic units
The volume of the solid is $$ \frac{256}{5} \pi $$ cubic units.

Calculate the volume of the solid generated by revolving the region bounded by the curve and the line around the x-axis.

Upstudy AI Solution

Answer

Solution

1. Identify the Points of Intersection

2. Set Up the Washer Method Integral

3. Express the Integral

4. Compute the Integral

5. Calculate the Volume

Final Answer

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