Define Q as the region bounded by the function u(y)= y^2/4 +1 and v(y)=1 between y=2 and y=4. If Q is rotated around the y axis , what is the volume of the resulting solid? submit an exact answer in terms of pi.         V= ? units^3

To find the volume \( V \) of the solid formed by rotating the region \( Q \) around the \( y \)-axis, we use the method of cylindrical shells. The formula for the volume of a solid of revolution using cylindrical shells is given by:

\[
V = \int_{a}^{b} 2\pi \cdot r(y) \cdot h(y) \, dy
\]

Where:
- \( r(y) \) is the radius of the shell, which in this case is the distance from the y-axis (which is simply \( y \) itself).
- \( h(y) \) is the height of the shell, which is the difference between the two functions \( u(y) - v(y) \).

Given:
- \( u(y) = \frac{y^2}{4} + 1 \)
- \( v(y) = 1 \)

The height \( h(y) \) is therefore:
\[
h(y) = u(y) - v(y) = \left(\frac{y^2}{4} + 1\right) - 1 = \frac{y^2}{4}
\]

Now, we calculate the volume using the integral from \( y = 2 \) to \( y = 4 \):

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

Simplify the integrand:

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

Now, compute the integral:

\[
\int y^3 \, dy = \frac{y^4}{4}
\]

Evaluate this from \( y = 2 \) to \( y = 4 \):

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

Calculate the values:

\[
4^4 = 256, \quad 2^4 = 16
\]

\[
V = \frac{\pi}{2} \left[\frac{256}{4} - \frac{16}{4}\right] = \frac{\pi}{2} \left[64 - 4\right] = \frac{\pi}{2} \cdot 60 = 30\pi
\]

Thus, the exact volume \( V \) is:

\[
V = 30\pi \, \text{units}^3
\]

Would you like more details or have any questions?

Here are some related questions:
1. How would the volume change if the region were rotated around the x-axis instead?
2. What is the volume if the bounds were different, say \( y = 1 \) to \( y = 3 \)?
3. How would you set up the integral if the functions were \( u(y) = \frac{y^3}{3} + 2 \) and \( v(y) = 2 \)?
4. How does the method of disks differ from the method of shells in this context?
5. How can you determine the centroid of the region \( Q \) before rotation?

**Tip:** When solving volumes of revolution, always visualize the problem and clearly distinguish between the radius and height functions.

Learn how to calculate the volume of a solid formed by rotating a region bounded by the functions u(y) = y^2/4 + 1 and v(y) = 1 around the y-axis. Detailed step-by-step solution using the cylindrical shells method. Submit an exact answer in terms of pi.

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