find the volume of the solid formed by revolving the region bounded by $y=\sqrt x, x=1, x=4$ about the $y$-axis using the disk method,

To find the volume of the solid formed by revolving the region bounded by $ y = \sqrt{x} $, $ x = 1 $, and $ x = 4 $ about the $ y $-axis using the disk method, we proceed as follows:

1. **Rewrite the equation $ y = \sqrt{x} $ in terms of $ x $:**
   \[
   x = y^2
   \]

2. **Identify the bounds for $ y $:**
   Since $ x = 1 $ and $ x = 4 $, we need to find the corresponding $ y $-values.
   - When $ x = 1 $: $ y = \sqrt{1} = 1 $.
   - When $ x = 4 $: $ y = \sqrt{4} = 2 $.
   
   Thus, the region we are revolving is between $ y = 1 $ and $ y = 2 $.

3. **Set up the volume integral using the disk method:**
   When revolving around the $ y $-axis, the radius of a disk at height $ y $ is the distance from the $ y $-axis to the curve, which is $ x = y^2 $.
   
   The area of each circular disk is:
   \[
   A(y) = \pi \cdot (\text{radius})^2 = \pi \cdot (y^2)^2 = \pi y^4
   \]

4. **Integrate to find the volume:**
   \[
   V = \int_{y=1}^{y=2} A(y) \, dy = \int_{1}^{2} \pi y^4 \, dy
   \]
   Factor out $ \pi $:
   \[
   V = \pi \int_{1}^{2} y^4 \, dy
   \]

5. **Evaluate the integral:**
   \[
   V = \pi \left[ \frac{y^5}{5} \right]_{1}^{2}
   \]
   Substitute the bounds:
   \[
   V = \pi \left( \frac{2^5}{5} - \frac{1^5}{5} \right) = \pi \left( \frac{32}{5} - \frac{1}{5} \right)
   \]
   Simplify:
   \[
   V = \pi \cdot \frac{31}{5} = \frac{31\pi}{5}
   \]

Thus, the volume of the solid is:
\[
V = \frac{31\pi}{5}
\]

Would you like a more detailed breakdown, or do you have further questions?

---

1. What if the region were revolved around the $ x $-axis instead?
2. How would the method change if we used the shell method for this problem?
3. What would be the result if the bounds changed to $ x = 0 $ and $ x = 4 $?
4. How does the disk method compare to the washer method in setup?
5. What happens to the volume if we change the function to $ y = x^2 $?

**Tip:** When setting up volume integrals, always check if rewriting the function in terms of the variable of integration simplifies the problem.

Find the volume of the solid formed by revolving the region bounded by $y=\sqrt x$, $x=1$, $x=4$ about the $y$-axis using the disk method.

Calculate the volume of a solid formed by revolving the region bounded by $y=\sqrt{x}$, $x=1$, and $x=4$ around the y-axis using the disk method. This solution includes step-by-step calculus techniques suitable for college-level students studying solids of revolution.

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