the curve ysquared =4-x is rotated through 360 degrees about the y axis determine the volume of the solid formed

The problem involves finding the volume of the solid generated by rotating the curve \( y^2 = 4 - x \) about the y-axis through 360 degrees. This is a standard problem of solids of revolution.

We can use the method of cylindrical shells, given the curve is in the form \( x \) as a function of \( y \).

### Step-by-step Solution:

1. **Express \( x \) in terms of \( y \)**:  
   From the given equation \( y^2 = 4 - x \), solve for \( x \):
   \[
   x = 4 - y^2
   \]

2. **Identify the bounds for \( y \)**:  
   Since the curve is symmetric about the y-axis, and \( y^2 = 4 - x \), the bounds for \( y \) are from \( y = -2 \) to \( y = 2 \) (because \( x = 0 \) when \( y = \pm 2 \)).

3. **Use the formula for the volume of a solid generated by rotating around the y-axis** (cylindrical shell method):
   The volume \( V \) is given by:
   \[
   V = 2\pi \int_{y_{\text{min}}}^{y_{\text{max}}} x(y) \cdot y \, dy
   \]
   where \( x(y) = 4 - y^2 \).

4. **Set up the integral**:
   \[
   V = 2\pi \int_{-2}^{2} (4 - y^2) \cdot y \, dy
   \]

5. **Simplify the integral**:  
   Since the integrand is odd (it involves \( y \cdot y^2 \), which is an odd function), the integral over symmetric limits (from \( -2 \) to \( 2 \)) will cancel out to 0. Therefore, we can integrate from 0 to 2 and double the result:
   \[
   V = 2 \times 2\pi \int_0^{2} (4 - y^2) \cdot y \, dy
   \]
   Simplifying:
   \[
   V = 4\pi \int_0^{2} (4y - y^3) \, dy
   \]

6. **Compute the integral**:
   \[
   V = 4\pi \left[ \frac{4y^2}{2} - \frac{y^4}{4} \right]_0^2
   \]
   Evaluate at \( y = 2 \):
   \[
   V = 4\pi \left[ \frac{4(2)^2}{2} - \frac{(2)^4}{4} \right]
   = 4\pi \left[ \frac{16}{2} - \frac{16}{4} \right]
   = 4\pi \left[ 8 - 4 \right]
   = 4\pi \times 4 = 16\pi
   \]

### Final Answer:
The volume of the solid formed by rotating the curve \( y^2 = 4 - x \) about the y-axis is \( \boxed{16\pi} \) cubic units.

Would you like more details or have any questions?

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Here are five related questions:
1. How would the method change if the curve were rotated around the x-axis?
2. What if the curve \( y^2 = 4 - x \) was rotated around a different axis, like \( x = 2 \)?
3. How can we solve for the volume using the disk method instead?
4. What if the curve was only rotated 180 degrees instead of 360?
5. How does the volume change if the curve is modified to \( y^2 = 9 - x \)?

**Tip**: When solving solids of revolution problems, always visualize the region and axis of rotation to choose the best method (disks, washers, or cylindrical shells).

The curve y^2 = 4 - x is rotated through 360 degrees about the y-axis. Determine the volume of the solid formed.

Learn how to calculate the volume of a solid formed by rotating the curve y^2 = 4 - x about the y-axis through 360 degrees. This problem covers key calculus concepts, including the cylindrical shell method and integration, and is suitable for advanced high school or early university students.

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