3. Find the volume of the region in the first octant between the sphere of radius 1 centered at the origin and the sphere of radius 9 centered at the origin two ways: (a) Set up the integral as one triple integral in spherical coordinates and evaluate. (b) Use geometry.
3. Find the volume of the region in the first octant between the sphere of radius 1 centered at the origin and the sphere of radius 9 centered at the origin two ways: (a) Set up the integral as one triple integral in spherical coordinates and evaluate. (b) Use geometry.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 3:**
Find the volume of the region in the first octant between the sphere of radius 1 centered at the origin and the sphere of radius 9 centered at the origin two ways:
(a) Set up the integral as one triple integral in spherical coordinates and evaluate.
(b) Use geometry.
**Explanation:**
This problem asks for the calculation of a volume in the first octant (the region where \(x, y, z \geq 0\)) that lies between two spheres: a smaller sphere with radius 1 and a larger sphere with radius 9, both centered at the origin.
**Approach (a): Spherical Coordinates Integration**
- **Spherical Coordinates**: The volume \(V\) can be found using integration in spherical coordinates \( (r, \theta, \phi) \) where:
- \( r \) is the radius,
- \( \theta \) is the azimuthal angle in the \(xy\)-plane from the x-axis,
- \( \phi \) is the polar angle from the z-axis.
- **Integration Limits**:
- \( r \) ranges from 1 to 9.
- \(\theta\) ranges from \(0\) to \(\frac{\pi}{2}\) (since it’s the first octant).
- \(\phi\) ranges from \(0\) to \(\frac{\pi}{2}\) (since it’s the first octant).
- **Integral Setup**:
\[
V = \int_{0}^{\frac{\pi}{2}} \int_{0}^{\frac{\pi}{2}} \int_{1}^{9} r^2 \sin \phi \, dr \, d\phi \, d\theta
\]
**Approach (b): Geometry**
The volume can also be found by direct geometric methods, understanding that the volume of a sphere sector in the first octant (an eighth of a sphere) can be calculated, and then the volume of the sphere with radius 1 can be subtracted from the sphere with radius 9.
- **Volume of a Sphere**:
- Volume of the full sphere is \((\frac{4}{3} \pi r^3)\).
- Volume of the first octant is \(\frac{1}{8} \times \frac{4}{3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F763773a4-b59e-428b-8a3f-bc3dd5fb97d6%2F8d45aa73-21a0-4902-8919-eb2dca76e0d2%2F4ts89nb_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 3:**
Find the volume of the region in the first octant between the sphere of radius 1 centered at the origin and the sphere of radius 9 centered at the origin two ways:
(a) Set up the integral as one triple integral in spherical coordinates and evaluate.
(b) Use geometry.
**Explanation:**
This problem asks for the calculation of a volume in the first octant (the region where \(x, y, z \geq 0\)) that lies between two spheres: a smaller sphere with radius 1 and a larger sphere with radius 9, both centered at the origin.
**Approach (a): Spherical Coordinates Integration**
- **Spherical Coordinates**: The volume \(V\) can be found using integration in spherical coordinates \( (r, \theta, \phi) \) where:
- \( r \) is the radius,
- \( \theta \) is the azimuthal angle in the \(xy\)-plane from the x-axis,
- \( \phi \) is the polar angle from the z-axis.
- **Integration Limits**:
- \( r \) ranges from 1 to 9.
- \(\theta\) ranges from \(0\) to \(\frac{\pi}{2}\) (since it’s the first octant).
- \(\phi\) ranges from \(0\) to \(\frac{\pi}{2}\) (since it’s the first octant).
- **Integral Setup**:
\[
V = \int_{0}^{\frac{\pi}{2}} \int_{0}^{\frac{\pi}{2}} \int_{1}^{9} r^2 \sin \phi \, dr \, d\phi \, d\theta
\]
**Approach (b): Geometry**
The volume can also be found by direct geometric methods, understanding that the volume of a sphere sector in the first octant (an eighth of a sphere) can be calculated, and then the volume of the sphere with radius 1 can be subtracted from the sphere with radius 9.
- **Volume of a Sphere**:
- Volume of the full sphere is \((\frac{4}{3} \pi r^3)\).
- Volume of the first octant is \(\frac{1}{8} \times \frac{4}{3
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