Find the volume of the solid obtained by rotating the region bounded by the given curves about the line x=3.    and  x=y^2, x=1



To find the volume of the solid obtained by rotating the region bounded by the curves \( x = y^2 \) and \( x = 1 \) around the line \( x = 3 \), we can use the **shell method**. 

### Step-by-step solution:

#### 1. Sketch the region:
- The curve \( x = y^2 \) is a parabola that opens to the right.
- The vertical line \( x = 1 \) is a boundary for the region.

The region bounded by these curves lies between \( y = -1 \) and \( y = 1 \), because at these points, \( y^2 = 1 \), giving \( x = 1 \).

#### 2. Shell Method Formula:
For a region rotated about a vertical line (like \( x = 3 \)), the volume using the shell method is given by:
\[
V = 2\pi \int_{y_{\text{min}}}^{y_{\text{max}}} ( \text{radius} ) ( \text{height} ) \, dy
\]
- **Radius**: The distance from the axis of rotation \( x = 3 \) to the curve \( x = y^2 \). This is \( 3 - x = 3 - y^2 \).
- **Height**: The height of the shell is the horizontal distance between the curve \( x = y^2 \) and the line \( x = 1 \), which is \( 1 - y^2 \).

#### 3. Setting up the integral:
- The limits of integration are from \( y = -1 \) to \( y = 1 \), since the region is symmetric around the y-axis.
- The formula becomes:
\[
V = 2\pi \int_{-1}^{1} (3 - y^2)(1 - y^2) \, dy
\]

#### 4. Simplifying the integrand:
Expand \( (3 - y^2)(1 - y^2) \):
\[
(3 - y^2)(1 - y^2) = 3(1 - y^2) - y^2(1 - y^2)
= 3 - 3y^2 - y^2 + y^4
= 3 - 4y^2 + y^4
\]

#### 5. Compute the integral:
Now, integrate term by term:
\[
V = 2\pi \int_{-1}^{1} (3 - 4y^2 + y^4) \, dy
\]
Since the integrand is an even function, we can double the integral from 0 to 1:
\[
V = 4\pi \int_0^{1} (3 - 4y^2 + y^4) \, dy
\]

#### 6. Evaluate the integral:
\[
\int_0^1 3 \, dy = 3y \Big|_0^1 = 3
\]
\[
\int_0^1 -4y^2 \, dy = -\frac{4y^3}{3} \Big|_0^1 = -\frac{4}{3}
\]
\[
\int_0^1 y^4 \, dy = \frac{y^5}{5} \Big|_0^1 = \frac{1}{5}
\]

#### 7. Add the results:
\[
V = 4\pi \left( 3 - \frac{4}{3} + \frac{1}{5} \right)
\]
Simplify the terms inside the parentheses:
\[
3 - \frac{4}{3} + \frac{1}{5} = \frac{45}{15} - \frac{20}{15} + \frac{3}{15} = \frac{28}{15}
\]

#### 8. Final answer:
\[
V = 4\pi \cdot \frac{28}{15} = \frac{112\pi}{15}
\]

The volume of the solid is \( \frac{112\pi}{15} \) cubic units.

---

Would you like more details on any part of this explanation, or do you have any questions?

Here are 5 related questions to explore further:
1. How do we apply the disk/washer method to solve similar problems?
2. What is the significance of symmetry in simplifying integrals?
3. How would the volume change if the rotation were around the y-axis instead of \( x = 3 \)?
4. Can the cylindrical shell method be applied to other curves, like circles?
5. How do we handle cases where the axis of rotation is outside the region?

**Tip:** Always check whether the function you're rotating is symmetric around the axis of rotation, as it can simplify the integration process!

Find the volume of the solid obtained by rotating the region bounded by the given curves about the line x=3. and x=y^2, x=1

Learn how to calculate the volume of a solid obtained by rotating the region bounded by the curves x=y^2 and x=1 about the line x=3. This problem explores the cylindrical shell method and covers integral calculus concepts with step-by-step explanations. Suitable for students in advanced calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation