Find the volume of the solid obtained by rotating the area between the graphs of y    =    x   2    and      x    =    2y   around the y-axis

To find the volume of the solid obtained by rotating the area between the graphs of \( y = x^2 \) and \( x = 2y \) around the y-axis, we can use the **method of cylindrical shells**.

### Step-by-step solution:

1. **Identify the curves:**
   - The curve \( y = x^2 \) represents a parabola.
   - The curve \( x = 2y \) is a straight line, which can be rewritten as \( y = \frac{x}{2} \).

2. **Determine the points of intersection:**
   To find the bounds for the integral, we need to find where these curves intersect by setting \( y = x^2 \) equal to \( y = \frac{x}{2} \):
   \[
   x^2 = \frac{x}{2}
   \]
   Multiply both sides by 2 to clear the fraction:
   \[
   2x^2 = x
   \]
   Rearrange this into standard form:
   \[
   2x^2 - x = 0
   \]
   Factor out \( x \):
   \[
   x(2x - 1) = 0
   \]
   So, \( x = 0 \) or \( x = \frac{1}{2} \).

3. **Set up the volume integral (Cylindrical shells method):**
   The volume \( V \) of the solid can be calculated by the formula:
   \[
   V = 2\pi \int_{x_1}^{x_2} x \cdot ( \text{height of shell} ) \, dx
   \]
   Here, the height of the shell is the difference in \( y \)-values between the curves: \( \frac{x}{2} - x^2 \).

4. **Find the bounds of integration:**
   The curves intersect at \( x = 0 \) and \( x = \frac{1}{2} \), so these will be our limits of integration.

5. **Set up the integral:**
   \[
   V = 2\pi \int_{0}^{\frac{1}{2}} x \left( \frac{x}{2} - x^2 \right) dx
   \]
   Simplify the expression inside the integral:
   \[
   V = 2\pi \int_{0}^{\frac{1}{2}} x \left( \frac{x}{2} - x^2 \right) dx = 2\pi \int_{0}^{\frac{1}{2}} \left( \frac{x^2}{2} - x^3 \right) dx
   \]

6. **Compute the integral:**
   \[
   V = 2\pi \left[ \frac{x^3}{6} - \frac{x^4}{4} \right]_0^{\frac{1}{2}}
   \]
   Now, evaluate the bounds:
   \[
   V = 2\pi \left( \left( \frac{\left(\frac{1}{2}\right)^3}{6} - \frac{\left(\frac{1}{2}\right)^4}{4} \right) - \left( 0 \right) \right)
   \]
   Simplify the terms:
   \[
   V = 2\pi \left( \frac{1}{48} - \frac{1}{64} \right)
   \]
   Find a common denominator (LCM of 48 and 64 is 192):
   \[
   V = 2\pi \left( \frac{4}{192} - \frac{3}{192} \right) = 2\pi \left( \frac{1}{192} \right)
   \]
   \[
   V = \frac{2\pi}{192} = \frac{\pi}{96}
   \]

### Final Answer:
The volume of the solid is \( \frac{\pi}{96} \) cubic units.

---

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

Here are 5 related questions to deepen your understanding:
1. How do you apply the cylindrical shells method when rotating around the x-axis instead?
2. How would the solution change if the rotation was around a vertical line other than the y-axis?
3. Can you solve the problem using the disk/washer method instead of cylindrical shells?
4. What is the geometric interpretation of the height of a cylindrical shell in this context?
5. How do you find points of intersection for more complex curves?

**Tip:** Always double-check the limits of integration when setting up the volume for rotated solids!

Find the volume of the solid obtained by rotating the area between the graphs of y = x^2 and x = 2y around the y-axis.

Learn how to find the volume of a solid obtained by rotating the area between y = x^2 and x = 2y around the y-axis using the cylindrical shells method. This problem covers key calculus concepts, including integration and solving intersections. Suitable for advanced high school students in Grades 11-12.

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