Find the volume of the solid obtained by rotating the region bounded by the curve \( y = \sqrt{x} \), the x-axis, and the line \( x = 4 \) about the y-axis.

To find the volume of the solid obtained by rotating the region bounded by the curve \( y = \sqrt{x} \), the x-axis, and the line \( x = 4 \) about the y-axis, we use the method of cylindrical shells.

### Step 1: Express \( x \) in terms of \( y \)
Given the curve \( y = \sqrt{x} \), we first solve for \( x \) in terms of \( y \):
\[
y = \sqrt{x} \implies x = y^2
\]

### Step 2: Set up the volume integral using the shell method
The formula for the volume of a solid of revolution using the method of cylindrical shells is:
\[
V = 2\pi \int_{a}^{b} ( \text{radius} ) \times (\text{height}) \, dy
\]
- **Radius**: The radius of a cylindrical shell is the distance from the axis of rotation (the y-axis), which is simply \( x = y^2 \).
- **Height**: The height of the cylindrical shell is given by the function \( x = y^2 \), ranging from \( y = 0 \) to \( y = 2 \) (since \( y = \sqrt{x} \) and \( x = 4 \)).

The volume integral becomes:
\[
V = 2\pi \int_{0}^{2} y \times y^2 \, dy
\]
\[
V = 2\pi \int_{0}^{2} y^3 \, dy
\]

### Step 3: Evaluate the integral
To evaluate the integral, first compute the antiderivative of \( y^3 \):
\[
\int y^3 \, dy = \frac{y^4}{4}
\]

Now, apply the limits of integration from 0 to 2:
\[
V = 2\pi \left[ \frac{y^4}{4} \right]_{0}^{2}
\]
\[
V = 2\pi \left( \frac{2^4}{4} - \frac{0^4}{4} \right)
\]
\[
V = 2\pi \left( \frac{16}{4} \right) = 2\pi \times 4 = 8\pi
\]

### Final Answer
The volume of the solid is \( \boxed{8\pi} \) cubic units.

Would you like more details or have any questions? 

Here are some related questions for further practice:
1. How would the volume change if the region were rotated about the x-axis instead?
2. What is the volume if the region is rotated about the line \( x = 6 \) instead of the y-axis?
3. How would you find the surface area of the solid formed by rotating the same region about the y-axis?
4. What if the curve was \( y = x^2 \) instead of \( y = \sqrt{x} \)? How would the volume change?
5. How does the volume calculation differ if the boundary is extended to \( x = 9 \)?

**Tip:** The shell method is particularly useful when the solid of revolution involves a function that is easier to express in terms of the radius and height relative to the axis of rotation.

Learn how to find the volume of a solid obtained by rotating the region bounded by y = sqrt(x), the x-axis, and x = 4 about the y-axis using the cylindrical shells method. Detailed step-by-step solution with integral calculus and formulas 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