The volume Bounded by x^2+y^2=9 and y^2+z^2=9 is

To find the volume bounded by the two surfaces \( x^2 + y^2 = 9 \) and \( y^2 + z^2 = 9 \), we need to interpret the geometry of these surfaces and compute the volume they enclose.

### Step 1: Understand the surfaces
1. The equation \( x^2 + y^2 = 9 \) represents a **cylinder** of radius 3 centered on the z-axis and extending infinitely along the z-axis. This cylinder is aligned along the z-axis.

2. The equation \( y^2 + z^2 = 9 \) represents another **cylinder**, but this one is aligned along the x-axis. It is also of radius 3, centered along the x-axis.

The volume we are interested in is the region where these two cylinders intersect. This type of problem is commonly referred to as the **intersection of two cylinders** or the **Steinmetz solid**.

### Step 2: Symmetry of the region
By symmetry, the region of intersection is symmetric in all directions. Therefore, we can compute the volume for a single quadrant (1/4th of the region) and multiply it by 4 to get the total volume.

### Step 3: Setup the integral
We will use a double integral to calculate the volume. First, convert to polar coordinates for easier integration.

- The intersection region can be bounded by \( x^2 + y^2 \leq 9 \) (circular disk of radius 3 in the xy-plane), but for polar coordinates, \( x = r \cos \theta \) and \( y = r \sin \theta \).
- The limits for \( r \) go from 0 to 3, and \( \theta \) will go from 0 to \( 2\pi \).

Using polar coordinates, the volume integral becomes:

\[
V = 4 \times \int_0^{\pi/2} \int_0^3 ( \text{upper limit for z} - \text{lower limit for z} ) \, r \, dr \, d\theta
\]

For each \( y = r \sin \theta \), \( z \) ranges from \( -\sqrt{9 - y^2} \) to \( \sqrt{9 - y^2} \), or equivalently:

\[
V = 4 \times \int_0^{\pi/2} \int_0^3 2\sqrt{9 - r^2} \, r \, dr \, d\theta
\]

### Step 4: Evaluate the integral

Let's first evaluate the inner integral:

\[
\int_0^3 2r \sqrt{9 - r^2} \, dr
\]

Using the substitution \( u = 9 - r^2 \), this evaluates to:

\[
V = 4 \times \frac{\pi}{2} \times \frac{9\pi}{2} = 36
\]

Thus, the total volume of the region is **36 cubic units**.

Would you like more details or explanations?

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Here are 5 related questions to expand your understanding:

1. How do we compute the volume of a region bounded by different types of surfaces?
2. What is the significance of the Steinmetz solid in geometry?
3. How can we extend this concept to higher-dimensional intersections?
4. What role does symmetry play in simplifying volume calculations?
5. How would you compute the volume if the cylinders had different radii?

**Tip:** Always take advantage of symmetry when calculating volumes for complex 3D shapes—it can greatly simplify your work!

The volume Bounded by x^2 + y^2 = 9 and y^2 + z^2 = 9 is

Learn how to calculate the volume of the region bounded by two intersecting cylinders using double integration and cylindrical coordinates. This step-by-step solution covers concepts like the Steinmetz solid, symmetry, and polar coordinate transformations. Suitable for advanced calculus students.

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