Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves 
y = 
4
x2 + 4
,
 y = 0, x = 0, and x = 2.

To find the volume of the solid obtained by rotating the region enclosed by the curves around the x-axis, we can use the **Disk Method**. The formula for the volume using the disk method when rotating about the x-axis is:

\[
V = \pi \int_{a}^{b} [f(x)]^2 \, dx
\]

### Step 1: Identify the curves and boundaries.
- The curve is \( y = \frac{4}{x^2 + 4} \).
- The boundaries are \( y = 0 \), \( x = 0 \), and \( x = 2 \).

### Step 2: Set up the integral.
Here, the function to rotate is \( f(x) = \frac{4}{x^2 + 4} \), and we are rotating it around the x-axis from \( x = 0 \) to \( x = 2 \).

The volume \( V \) is given by:
\[
V = \pi \int_{0}^{2} \left( \frac{4}{x^2 + 4} \right)^2 \, dx
\]

### Step 3: Simplify the integrand.
First, let's square the function inside the integral:
\[
V = \pi \int_{0}^{2} \left( \frac{16}{(x^2 + 4)^2} \right) \, dx
\]

### Step 4: Solve the integral.
To solve the integral, you can apply substitution methods or refer to standard integral tables, as it is a complex rational function. The integral of this specific form leads to:

\[
V = \pi \left[ \frac{x}{x^2 + 4} + \frac{1}{2} \arctan\left(\frac{x}{2}\right) \right]_{0}^{2}
\]

### Step 5: Evaluate the integral at the limits.
Now, plug in the limits \( x = 2 \) and \( x = 0 \) into the evaluated antiderivative:

At \( x = 2 \):
\[
\frac{2}{2^2 + 4} + \frac{1}{2} \arctan\left(\frac{2}{2}\right) = \frac{2}{8} + \frac{1}{2} \arctan(1) = \frac{1}{4} + \frac{\pi}{8}
\]

At \( x = 0 \):
\[
\frac{0}{0^2 + 4} + \frac{1}{2} \arctan\left(\frac{0}{2}\right) = 0
\]

### Step 6: Subtract and multiply by \( \pi \).
\[
V = \pi \left( \left( \frac{1}{4} + \frac{\pi}{8} \right) - 0 \right)
\]

\[
V = \pi \left( \frac{1}{4} + \frac{\pi}{8} \right)
\]

\[
V = \frac{\pi}{4} + \frac{\pi^2}{8}
\]

Thus, the volume of the solid is:

\[
V = \frac{\pi}{4} + \frac{\pi^2}{8} \, \text{cubic units}.
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the Disk Method differ from the Shell Method in volume calculations?
2. What is the geometric interpretation of revolving a region around an axis?
3. Can you apply this method to any bounded region, or are there restrictions?
4. How would the volume change if the region were revolved around the y-axis instead?
5. How would you handle more complex curves or boundaries in rotation problems?

**Tip:** Understanding the limits of integration is crucial when setting up volume problems involving rotation, as they define the boundaries of the region being revolved.

Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves y = 4/(x^2 + 4), y = 0, x = 0, and x = 2.

This page explains how to find the volume of a solid formed by rotating the region bounded by y = 4/(x² + 4), y = 0, x = 0, and x = 2 around the x-axis using the Disk Method. The solution involves integrating and applying calculus concepts, suitable for undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation