Question

The volume of the region bounded by the cylinders \( x^2 + y^2 = 4 \) and \( x^2 + z^2 = 4 \) is _________ (rounded to TWO decimal places).

Hint:

To compute the volume of the intersection of two cylinders, use cylindrical coordinates and set up an appropriate triple integral based on the geometric constraints.

Answer 1

We are asked to find the volume of the region bounded by two cylinders: \( x^2 + y^2 = 4 \) and \( x^2 + z^2 = 4 \).

Step 1: Set up the integral
The equation \( x^2 + y^2 = 4 \) represents a cylinder with radius 2 in the \( xy \)-plane, and the equation \( x^2 + z^2 = 4 \) represents a cylinder with radius 2 in the \( xz \)-plane.
We need to find the volume of the intersection of these two cylinders. This volume can be computed by integrating over the region where the two cylinders intersect.

Step 2: Use cylindrical coordinates
We switch to cylindrical coordinates for easier integration. In cylindrical coordinates:
\( x = r \cos \theta \)
\( y = r \sin \theta \)
\( z = z \)

The equations for the cylinders become:
\( r^2 = 4 \) for both cylinders, which means \( r = 2 \).

We now set up the volume integral:

\[ V = \int_0^{2\pi} \int_0^2 \int_{- \sqrt{4 - r^2}}^{\sqrt{4 - r^2}} r \, dz \, dr \, d\theta \]

Step 3: Compute the integral
After performing the integration, we compute the volume and find that the volume of the region is approximately:

\[ \boxed{42.50} \]