https://file.aiquickdraw.com/mathbot/file/1725508571170q1cept2c.jpg

The problem in the image asks for the volume of the solid generated by revolving the region bounded by the curves:

- \( 4x = y^2 \)
- \( 10x = y = 10 \)
- \( x = 0 \)

about the **y-axis**.

This is a typical problem involving the use of the *shell method* for finding the volume of solids of revolution. Here's how we can solve it:

### Step 1: Sketching the Region
The curves describe a parabola and two vertical lines:
1. \( 4x = y^2 \) is a sideways parabola with vertex at \( (0,0) \).
2. \( 10x = y = 10 \) suggests the line \( x = \frac{10}{10} = 1 \) as the upper boundary for \( y = 10 \).
3. \( x = 0 \) is the y-axis.

Thus, the region lies between these curves.

### Step 2: Setting up the Shell Method Formula
For the shell method, the formula for volume is:

\[
V = 2\pi \int_{y_{\text{min}}}^{y_{\text{max}}} (radius)(height) \, dy
\]

Here:
- The **radius** is the distance from the y-axis, which is simply \( x(y) \).
- The **height** is given by \( x(y) = \frac{y^2}{4} \).

Thus, the formula becomes:

\[
V = 2\pi \int_{0}^{10} y \cdot \frac{y^2}{4} \, dy
\]

### Step 3: Solving the Integral
Now, let's calculate the integral.

\[
V = 2\pi \int_{0}^{10} \frac{y^3}{4} \, dy
\]

\[
V = \frac{\pi}{2} \left[ \frac{y^4}{4} \right]_{0}^{10}
\]

\[
V = \frac{\pi}{2} \left[ \frac{10^4}{4} - 0 \right]
\]

\[
V = \frac{\pi}{2} \times \frac{10000}{4} = \frac{\pi}{2} \times 2500 = 1250\pi
\]

### Step 4: Conclusion
The volume of the solid generated by revolving the region is **1250π**. Based on the options in the image, this matches **option E (None of the above)**, since none of the choices provided give \( 1250\pi \).

Would you like more details or have any further questions on this?

### Related Questions:
1. How is the shell method different from the disk/washer method?
2. How do you determine which method (shell vs. disk) to use for solids of revolution?
3. Can you explain the process of changing variables when revolving around axes other than the x-axis or y-axis?
4. What happens if the region is revolved around a line not on an axis, such as \( y = 2 \)?
5. How would the integral change if the parabola \( 4x = y^2 \) opened upwards instead of sideways?

**Tip**: Always double-check the limits of integration when setting up your integral to avoid mistakes in the volume calculation.

Sketch the region bounded by the curves 4x = y^2, 10x = y = 10, and x = 0. Find the volume of the solid generated by revolving the region about the y-axis.

Learn how to calculate the volume of a solid generated by revolving the region bounded by 4x = y^2, 10x = y = 10, and x = 0 around the y-axis. This problem involves using the shell method to find the volume, making it ideal for Grade 12 or University Calculus students.

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