te /// (+ (xy + yz + xz)dV B B = {(x, y, z) | 0≤x≤ 10,0 ≤ y ≤ 7,0 < z < 1}. Evaluate
te /// (+ (xy + yz + xz)dV B B = {(x, y, z) | 0≤x≤ 10,0 ≤ y ≤ 7,0 < z < 1}. Evaluate
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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![**Topic: Evaluating Triple Integrals Over a Rectangular Region**
**Problem:**
Evaluate the triple integral:
\[
\iiint\limits_B (xy + yz + xz) \, dV
\]
where the region \( B \) is defined as:
\[
B = \{(x, y, z) \,|\, 0 \leq x \leq 10, \, 0 \leq y \leq 7, \, 0 \leq z \leq 1\} .
\]
**Explanation:**
This problem involves evaluating a triple integral over a specified rectangular region in three-dimensional space. The integrand is a linear combination of the product of the variables \( x \), \( y \), and \( z \).
**Steps to Solve:**
1. **Identify the limits of integration:**
- \( x \) ranges from 0 to 10.
- \( y \) ranges from 0 to 7.
- \( z \) ranges from 0 to 1.
2. **Set up the integral:**
The integral will be evaluated iteratively, typically in the order of \( dz \), \( dy \), and \( dx \), unless specified otherwise.
3. **Evaluate the inner integral first:**
- Integrate with respect to \( z \) from 0 to 1.
4. **Evaluate the middle integral:**
- Integrate the result with respect to \( y \) from 0 to 7.
5. **Evaluate the outer integral:**
- Integrate the result with respect to \( x \) from 0 to 10.
By following these steps, you will obtain the value of the triple integral over the region \( B \). This is a common method for solving problems involving volume integrals in multivariable calculus and is useful in applications such as computing mass, charge, or other quantities distributed in a three-dimensional space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd20dfe5a-a4c1-4793-9f05-a80ad59a67d4%2Faa0f2a3e-8b67-418c-a9f8-456df9402b79%2F1a8ppg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Evaluating Triple Integrals Over a Rectangular Region**
**Problem:**
Evaluate the triple integral:
\[
\iiint\limits_B (xy + yz + xz) \, dV
\]
where the region \( B \) is defined as:
\[
B = \{(x, y, z) \,|\, 0 \leq x \leq 10, \, 0 \leq y \leq 7, \, 0 \leq z \leq 1\} .
\]
**Explanation:**
This problem involves evaluating a triple integral over a specified rectangular region in three-dimensional space. The integrand is a linear combination of the product of the variables \( x \), \( y \), and \( z \).
**Steps to Solve:**
1. **Identify the limits of integration:**
- \( x \) ranges from 0 to 10.
- \( y \) ranges from 0 to 7.
- \( z \) ranges from 0 to 1.
2. **Set up the integral:**
The integral will be evaluated iteratively, typically in the order of \( dz \), \( dy \), and \( dx \), unless specified otherwise.
3. **Evaluate the inner integral first:**
- Integrate with respect to \( z \) from 0 to 1.
4. **Evaluate the middle integral:**
- Integrate the result with respect to \( y \) from 0 to 7.
5. **Evaluate the outer integral:**
- Integrate the result with respect to \( x \) from 0 to 10.
By following these steps, you will obtain the value of the triple integral over the region \( B \). This is a common method for solving problems involving volume integrals in multivariable calculus and is useful in applications such as computing mass, charge, or other quantities distributed in a three-dimensional space.
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