To find the volume of the solid formed when the given region is revolved about the y-axis, we will use the method of cylindrical shells. The equation of the curve is:

$f(x) = x^2 \sqrt{16 - x^2}$

We are rotating the region bounded by this function and the x-axis over the interval $0 \leq x \leq 4$.

Step-by-Step Solution Using the Method of Cylindrical Shells:

The volume of a solid of revolution using the method of cylindrical shells is given by:

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

Here, $x$ represents the radius of a typical shell, $f(x)$ is the height of the shell, and $[a, b]$ is the interval over which we are integrating.

Step 1: Identify the bounds and function

• The function is $f(x) = x^2 \sqrt{16 - x^2}$.
• The bounds are $a = 0$ and $b = 4$.

Step 2: Set up the integral

We will now set up the integral for the volume:

$V = 2\pi \int_{0}^{4} x \left( x^2 \sqrt{16 - x^2} \right) dx$

Simplifying the integrand:

$V = 2\pi \int_{0}^{4} x^3 \sqrt{16 - x^2} \, dx$

Step 3: Substitution for the integral

Let $u = 16 - x^2$, then $du = -2x \, dx$.

This substitution simplifies the integral:

• When $x = 0$, $u = 16$.
• When $x = 4$, $u = 0$.

Thus, the integral becomes:

$V = 2\pi \int_{16}^{0} -\frac{1}{2} (16 - u) \sqrt{u} \, du$

We can now calculate this integral to find the volume of the solid.

I will solve the integral now.The volume of the solid formed by revolving the region around the y-axis is:

$V = \frac{4096\pi}{15} \text{ cubic units}.$

This is the exact value of the volume.

Would you like further clarification or details?

Here are 5 related questions to expand on this:

1. How does the method of cylindrical shells differ from the disk/washer method for finding volumes?
2. What is the physical meaning of the term $2\pi \int x \cdot f(x) \, dx$ in this context?
3. How can you apply a similar method to revolve the region around the x-axis instead?
4. How would the integral change if the region were rotated around a different line, such as $x = 2$?
5. How can you visualize the solid created by this revolution?

Tip: Always check if your function is symmetric when applying integration techniques. Symmetry can simplify the calculation.