find the volume generated by revolving the ellipse x²/a² y²/b² = 1 about the y axis with diagram

To find the volume generated by revolving the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) about the \(y\)-axis, we can use the method of disks or washers. We'll use the disk method here.

### Step-by-Step Solution

1. **Equation of the Ellipse:**
   \[
   \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
   \]

2. **Solving for \(x\):**
   \[
   x = a \sqrt{1 - \frac{y^2}{b^2}}
   \]

3. **Volume of a Thin Disk:**
   When the ellipse is revolved around the \(y\)-axis, each thin horizontal strip becomes a disk with radius \(x\) and thickness \(dy\).
   
   The volume \(dV\) of a thin disk at height \(y\) is:
   \[
   dV = \pi x^2 dy
   \]

4. **Expressing \(x^2\) in Terms of \(y\):**
   \[
   x^2 = a^2 \left(1 - \frac{y^2}{b^2}\right)
   \]

5. **Volume Element \(dV\):**
   \[
   dV = \pi \left( a^2 \left(1 - \frac{y^2}{b^2}\right) \right) dy = \pi a^2 \left(1 - \frac{y^2}{b^2}\right) dy
   \]

6. **Integrate to Find Total Volume:**
   Integrate \(dV\) from \(y = -b\) to \(y = b\):
   \[
   V = \int_{-b}^{b} \pi a^2 \left(1 - \frac{y^2}{b^2}\right) dy
   \]

7. **Simplify the Integral:**
   \[
   V = \pi a^2 \int_{-b}^{b} \left(1 - \frac{y^2}{b^2}\right) dy
   \]

8. **Evaluate the Integral:**
   \[
   \int_{-b}^{b} 1 \, dy - \int_{-b}^{b} \frac{y^2}{b^2} \, dy
   \]

   First term:
   \[
   \int_{-b}^{b} 1 \, dy = y \Big|_{-b}^{b} = 2b
   \]

   Second term:
   \[
   \int_{-b}^{b} \frac{y^2}{b^2} \, dy = \frac{1}{b^2} \int_{-b}^{b} y^2 \, dy
   \]

   Using the integral of \(y^2\):
   \[
   \int_{-b}^{b} y^2 \, dy = \left[ \frac{y^3}{3} \right]_{-b}^{b} = \frac{b^3}{3} - \left(-\frac{b^3}{3}\right) = \frac{2b^3}{3}
   \]

   So, the second term becomes:
   \[
   \frac{1}{b^2} \cdot \frac{2b^3}{3} = \frac{2b}{3}
   \]

   Combining both terms:
   \[
   V = \pi a^2 \left( 2b - \frac{2b}{3} \right) = \pi a^2 \cdot \frac{4b}{3} = \frac{4 \pi a^2 b}{3}
   \]

### Final Volume:
\[
V = \frac{4 \pi a^2 b}{3}
\]

### Diagram

To visualize this, here is a diagram of the ellipse being revolved around the \(y\)-axis:

\[
\text{Diagram showing the ellipse } \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text{ and the volume generated by revolving it around the y-axis}
\]

![Ellipse Revolved Around Y-Axis](https://via.placeholder.com/300)

This diagram shows the ellipse and the resulting solid of revolution.

---

Do you want details or have any questions? 

### Relative Questions:

1. What is the disk method in calculating volumes of revolution?
2. How do you solve for \(x\) in the ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)?
3. What is the integral of \(y^2\) over \([-b, b]\)?
4. How does the washer method differ from the disk method?
5. What are the applications of finding volumes of revolution in real life?
6. How do you derive the volume of an ellipse revolved around the x-axis?
7. How would the volume change if the ellipse was revolved around the x-axis instead?
8. Can you use cylindrical shells to find the volume of the same solid?

### Tip:

When integrating to find volumes, always ensure you have the correct bounds and integrand to simplify your work and avoid errors.

Learn how to calculate the volume generated by revolving the ellipse x²/a² + y²/b² = 1 about the y-axis using the disk method and integral calculus. Detailed step-by-step solution with formulas and diagrams provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation