Find the volume of the solid region F.
The region F is the region in the first octant that is bounded by the two parabolic cylinders 
z = 64 − y2
 and 
z = 64 − x2.

To find the volume of the solid region \( F \), we need to calculate the volume of the region in the first octant bounded by the parabolic cylinders \( z = 64 - y^2 \) and \( z = 64 - x^2 \).

### Step 1: Identify the bounds of the region

We are working in the first octant, so:

- \( x \geq 0 \)
- \( y \geq 0 \)
- \( z \geq 0 \)

The surfaces are defined as follows:

1. The surface \( z = 64 - y^2 \) is a parabolic cylinder parallel to the \( x \)-axis.
2. The surface \( z = 64 - x^2 \) is a parabolic cylinder parallel to the \( y \)-axis.

The region is enclosed where these two parabolic cylinders intersect.

### Step 2: Find the intersection curve

To find the intersection of the two surfaces, set \( z = 64 - y^2 \) equal to \( z = 64 - x^2 \):

\[
64 - y^2 = 64 - x^2
\]
\[
y^2 = x^2
\]
\[
y = x
\]

Thus, the intersection of the two surfaces in the first octant is along the line \( y = x \). 

### Step 3: Set up the volume integral

We will integrate in the order \( dz \), \( dy \), and \( dx \). The upper bound of \( z \) is determined by the parabolic surfaces \( z = 64 - y^2 \) or \( z = 64 - x^2 \).

For each slice in the \( xy \)-plane, the \( x \) and \( y \)-coordinates vary within the region bounded by \( y = 0 \) and \( y = x \), and \( x = 0 \) to \( x = 8 \) (since at \( x = 8 \), \( z = 0 \)).

Thus, the volume \( V \) can be expressed as:

\[
V = \int_0^8 \int_0^x \int_0^{64 - y^2} dz \, dy \, dx
\]

### Step 4: Evaluate the integral

First, integrate with respect to \( z \):

\[
\int_0^{64 - y^2} dz = (64 - y^2)
\]

Now the volume integral becomes:

\[
V = \int_0^8 \int_0^x (64 - y^2) \, dy \, dx
\]

### Step 5: Integrate with respect to \( y \)

\[
\int_0^x (64 - y^2) \, dy = \left[ 64y - \frac{y^3}{3} \right]_0^x = 64x - \frac{x^3}{3}
\]

Now the volume integral becomes:

\[
V = \int_0^8 \left( 64x - \frac{x^3}{3} \right) dx
\]

### Step 6: Integrate with respect to \( x \)

\[
\int_0^8 \left( 64x - \frac{x^3}{3} \right) dx = \left[ 32x^2 - \frac{x^4}{12} \right]_0^8
\]

Evaluate at the bounds \( x = 8 \) and \( x = 0 \):

\[
V = \left( 32(8)^2 - \frac{(8)^4}{12} \right) - 0
\]
\[
V = \left( 32(64) - \frac{4096}{12} \right)
\]
\[
V = (2048 - 341.33) = 1706.67
\]

### Final Answer:
The volume of the solid region \( F \) is approximately \( 1706.67 \) cubic units.

---

Would you like further details or clarifications on any step?

### Related Questions:
1. How do you find the volume of a solid in cylindrical coordinates?
2. What is the geometric meaning of a parabolic cylinder?
3. How would the integral change if the region was not limited to the first octant?
4. Can this method be applied to other regions bounded by paraboloids?
5. How would you solve this problem using polar coordinates?

### Tip:
When dealing with volumes of solids, always check if symmetry can simplify the integration process. Symmetry often reduces the complexity of the limits of integration.

Find the volume of the solid region F. The region F is the region in the first octant that is bounded by the two parabolic cylinders z = 64 − y^2 and z = 64 − x^2.

Find the volume of the solid region in the first octant bounded by the parabolic cylinders z = 64 − y^2 and z = 64 − x^2 using triple integration. This step-by-step solution covers multivariable calculus concepts, including volume of solids and triple integrals. Suitable for undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation