Find the mass of the cylinder (centered on the z-axis and base in the xy-plane) with radius 4 and Z (set-up triple integral then evaluate by hand) x² + y² +1 height 6 and density function f(x, y, z) = - .
Find the mass of the cylinder (centered on the z-axis and base in the xy-plane) with radius 4 and Z (set-up triple integral then evaluate by hand) x² + y² +1 height 6 and density function f(x, y, z) = - .
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 Statement
**Objective:** Determine the mass of a cylinder.
**Specifications:**
- **Cylinder Dimensions:**
- **Radius:** 4
- **Height:** 6
- **Center:** On the z-axis
- **Base:** In the xy-plane
- **Density Function:**
\[
f(x, y, z) = \frac{z}{x^2 + y^2 + 1}
\]
- **Instructions:** Set up the triple integral to calculate the mass and then evaluate it by hand.
### Solution Approach
To find the mass of the given cylinder with the specified density function, follow these steps:
1. **Define the Region of Integration:**
The cylinder is centered on the z-axis with its base in the xy-plane:
- **For x and y:** The radius is 4, so the region in the xy-plane is a circle of radius 4.
- **For z:** The height ranges from 0 to 6.
2. **Set up the Triple Integral:**
The mass \( M \) of the cylinder is given by the integral of the density function over the volume of the cylinder:
\[
M = \int \int \int_V f(x, y, z) \, dV
\]
3. **Transform to Cylindrical Coordinates:**
To simplify the calculation, switch to cylindrical coordinates where:
\[
x = r \cos \theta, \, y = r \sin \theta, \, z = z
\]
And the volume element \( dV \) becomes:
\[
dV = r \, dr \, d\theta \, dz
\]
4. **Substitute and Integrate:**
Substitute the limits and the function \( f(x, y, z) \) expressed in cylindrical coordinates into the integral:
\[
x^2 + y^2 = r^2
\]
Then:
\[
f(r, \theta, z) = \frac{z}{r^2 + 1}
\]
Therefore, the integral becomes:
\[
M = \int_{0}^{2\pi} \int_{0}^{4} \int_{0}^{6} \frac{z}{r^2 + 1} \cdot](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff930c139-ad70-48e2-a90a-c5db80131496%2Ff7bcac00-f901-44ad-94e1-d1d0e7da9504%2Fjby28x2k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Objective:** Determine the mass of a cylinder.
**Specifications:**
- **Cylinder Dimensions:**
- **Radius:** 4
- **Height:** 6
- **Center:** On the z-axis
- **Base:** In the xy-plane
- **Density Function:**
\[
f(x, y, z) = \frac{z}{x^2 + y^2 + 1}
\]
- **Instructions:** Set up the triple integral to calculate the mass and then evaluate it by hand.
### Solution Approach
To find the mass of the given cylinder with the specified density function, follow these steps:
1. **Define the Region of Integration:**
The cylinder is centered on the z-axis with its base in the xy-plane:
- **For x and y:** The radius is 4, so the region in the xy-plane is a circle of radius 4.
- **For z:** The height ranges from 0 to 6.
2. **Set up the Triple Integral:**
The mass \( M \) of the cylinder is given by the integral of the density function over the volume of the cylinder:
\[
M = \int \int \int_V f(x, y, z) \, dV
\]
3. **Transform to Cylindrical Coordinates:**
To simplify the calculation, switch to cylindrical coordinates where:
\[
x = r \cos \theta, \, y = r \sin \theta, \, z = z
\]
And the volume element \( dV \) becomes:
\[
dV = r \, dr \, d\theta \, dz
\]
4. **Substitute and Integrate:**
Substitute the limits and the function \( f(x, y, z) \) expressed in cylindrical coordinates into the integral:
\[
x^2 + y^2 = r^2
\]
Then:
\[
f(r, \theta, z) = \frac{z}{r^2 + 1}
\]
Therefore, the integral becomes:
\[
M = \int_{0}^{2\pi} \int_{0}^{4} \int_{0}^{6} \frac{z}{r^2 + 1} \cdot
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