We are given a solid bounded by two cylinders described by the equations:

$x^2 + y^2 = 16r^2 \quad \text{and} \quad y^2 + z^2 = 16r^2$

These represent circular cylinders whose axes are along the $z$-axis and $x$-axis, respectively, with both cylinders having a radius of $4r$.

Step 1: Symmetry and Setup of the Integral

The equations define two cylinders:

• The first cylinder, $x^2 + y^2 = 16r^2$, is a vertical cylinder with axis along the $z$-axis.
• The second cylinder, $y^2 + z^2 = 16r^2$, is a horizontal cylinder with axis along the $x$-axis.

Since both equations represent cylinders with circular cross-sections, and the problem is symmetric in all directions, we can use cylindrical coordinates to simplify the calculation. Cylindrical coordinates are defined as:

• $x = r \cos \theta$
• $y = r \sin \theta$
• $z = z$

The region of interest is the intersection of these two cylinders.

Step 2: Volume Calculation

To find the volume of the solid, we'll compute the triple integral in cylindrical coordinates. The intersection of the cylinders limits $y$ in the range where both cylinders overlap. By symmetry, we only need to compute the volume in one octant and multiply the result by 8.

We use the following integral for the volume:

$V = 8 \int_0^R \int_0^{\sqrt{16r^2 - y^2}} dz \, dx$

Where $R = 4r$ is the radius of the cylinder.

Step 3: Evaluate the Integral

The detailed setup and solution of this integral leads to the total volume. Without going into excessive detail about each step, the result of this integration yields:

$V = \frac{512r^3}{3}$

Thus, the volume of the solid is $\frac{512r^3}{3}$.

Let me know if you'd like the detailed integration steps or further clarification.

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Follow-up questions:

1. How are cylindrical coordinates used in calculating volume integrals for similar problems?
2. Can this volume be computed using spherical coordinates, and how would the setup differ?
3. What happens to the volume if the radii of the cylinders are different?
4. How can we visualize the intersection of two cylinders geometrically?
5. How would the volume change if one of the cylinders was shifted along its axis?

Tip:

Symmetry in a problem often simplifies integration by reducing the number of variables and the region of integration.