To solve this problem using the method of cylindrical shells, let's first break it down into the key components and then answer the specific question about the circumference $c$ and height $h$ of a typical shell.

Volume of Solid Using Cylindrical Shells:

The formula for the volume $V$ of a solid of revolution obtained by rotating a region around the y-axis using cylindrical shells is:

$V = \int_{a}^{b} 2\pi x \cdot h(x) \, dx$

Where:

• $x$ is the radius of the shell (distance from the y-axis),
• $h(x)$ is the height of the shell (the function defining the region),
• $2\pi x$ is the circumference of the shell,
• $[a, b]$ is the interval for $x$.

Step 1: Circumference $c$

For the method of cylindrical shells, the radius of a typical shell is the $x$-coordinate of the point on the curve being rotated. The circumference of the shell is the distance around the shell, given by:

$c = 2\pi x$

Thus, the circumference of a typical cylindrical shell at $x$ is:

$c = 2\pi x$

Step 2: Height $h$

The height of a typical shell is the vertical distance between the curve $y = 4\sin(2x^2)$ and the x-axis (since we're rotating around the y-axis). Therefore, the height of the shell is simply the value of the function at that point, i.e., $h(x) = 4\sin(2x^2)$.

So, the height of a typical shell is:

$h(x) = 4\sin(2x^2)$

Step 3: Interval of Integration

Since we are given that $x = \sqrt{\frac{\pi}{2}}$ is the upper limit of $x$, the volume integral will be evaluated over the interval $[0, \sqrt{\frac{\pi}{2}}]$.

Let me know if you'd like to proceed with calculating the full volume or if you have any questions about these steps.

Here are 5 questions that extend the current problem:

1. How do you compute the integral of $4 \sin(2x^2)$?
2. What happens to the method of cylindrical shells when rotating around an axis other than the y-axis?
3. Can you use the disk method to solve this problem instead of cylindrical shells?
4. How do you determine the limits of integration when rotating around the x-axis?
5. What role does symmetry play in calculating volumes of solids of revolution?

Tip: Always identify the axis of rotation clearly before applying any integration method.