√√√₂ (4 B = {(x, y, z) | 0≤x≤ 3,0 ≤ y ≤ 6,0 ≤ z ≤ 7} . Evaluate (4z³ + 3y² + 2x)dV
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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Question
![**Example of Triple Integral Evaluation**
**Problem:**
Evaluate the triple integral:
\[
\iiint_B (4x^3 + 3y^2 + 2x) \, dV
\]
where \( B = \{(x, y, z) \mid 0 \leq x \leq 3, 0 \leq y \leq 6, 0 \leq z \leq 7\} \).
**Solution:**
To solve this problem, we are required to evaluate the triple integral of the function \(4x^3 + 3y^2 + 2x\) over the region \(B\), which is defined by the bounds:
- \(0 \leq x \leq 3\)
- \(0 \leq y \leq 6\)
- \(0 \leq z \leq 7\)
The region \(B\) is a rectangular box (or cuboid) in three-dimensional space. The integration will proceed in the order of integration, typically either \(dz\, dy\, dx\) or in any order that suits boundary conditions and simplifies computation.
1. **Define the bounds** for integration:
- The x-bound is from 0 to 3.
- The y-bound is from 0 to 6.
- The z-bound is from 0 to 7.
2. **Integrate the function** \( (4x^3 + 3y^2 + 2x) \) over these boundaries.
3. **Result interpretation**:
- The final result of the integration will give the accumulated value of the function \(4x^3 + 3y^2 + 2x\) over the defined volume \(B\).
This process involves computing nested definite integrals, which become computationally straightforward since the boundaries are constants and the region is rectangular.
**Notes:**
- The function \(4x^3 + 3y^2 + 2x\) is a polynomial expression, making it suitable for direct integration using basic rules of calculus.
- Each different permutation of integration sequences (like integrating with respect to \(z\) first, then \(y\), then \(x\)) might simplify the computation depending on the function involved, but due to the symmetries and independence of the bounds here, any order will yield the same result.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1f24c0c-e250-4c2d-b03e-5d045af5839c%2Fa3230908-2772-4c3e-a315-6929cf16e0d0%2Fl9sqz19_processed.png&w=3840&q=75)
Transcribed Image Text:**Example of Triple Integral Evaluation**
**Problem:**
Evaluate the triple integral:
\[
\iiint_B (4x^3 + 3y^2 + 2x) \, dV
\]
where \( B = \{(x, y, z) \mid 0 \leq x \leq 3, 0 \leq y \leq 6, 0 \leq z \leq 7\} \).
**Solution:**
To solve this problem, we are required to evaluate the triple integral of the function \(4x^3 + 3y^2 + 2x\) over the region \(B\), which is defined by the bounds:
- \(0 \leq x \leq 3\)
- \(0 \leq y \leq 6\)
- \(0 \leq z \leq 7\)
The region \(B\) is a rectangular box (or cuboid) in three-dimensional space. The integration will proceed in the order of integration, typically either \(dz\, dy\, dx\) or in any order that suits boundary conditions and simplifies computation.
1. **Define the bounds** for integration:
- The x-bound is from 0 to 3.
- The y-bound is from 0 to 6.
- The z-bound is from 0 to 7.
2. **Integrate the function** \( (4x^3 + 3y^2 + 2x) \) over these boundaries.
3. **Result interpretation**:
- The final result of the integration will give the accumulated value of the function \(4x^3 + 3y^2 + 2x\) over the defined volume \(B\).
This process involves computing nested definite integrals, which become computationally straightforward since the boundaries are constants and the region is rectangular.
**Notes:**
- The function \(4x^3 + 3y^2 + 2x\) is a polynomial expression, making it suitable for direct integration using basic rules of calculus.
- Each different permutation of integration sequences (like integrating with respect to \(z\) first, then \(y\), then \(x\)) might simplify the computation depending on the function involved, but due to the symmetries and independence of the bounds here, any order will yield the same result.
Expert Solution
Step 1
The given problem is to evaluate the given triple integral over the given regions of x,y,z.
Step by step
Solved in 2 steps with 1 images
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