Evaluate ₁ (42³ + 3y² + 2x)dV {(x, y, z) | 0≤x≤ 8,0 ≤ y ≤ 8,0 ≤ z ≤9}. B
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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![**Triple Integral Evaluation over a Rectangular Prism**
**Problem Statement:**
Evaluate the triple integral:
\[
\iiint\limits_{B} (4x^3 + 3y^2 + 2x) \, dV
\]
where the region \( B \) is defined as:
\[
B = \{ (x, y, z) \mid 0 \leq x \leq 8, \, 0 \leq y \leq 8, \, 0 \leq z \leq 9 \}
\]
**Region Description:**
The region \( B \) is a rectangular prism (cuboid) in 3-dimensional space. Its dimensions are given by the intervals on \( x \), \( y \), and \( z \):
- \( x \) ranges from 0 to 8.
- \( y \) ranges from 0 to 8.
- \( z \) ranges from 0 to 9.
The volume \( dV \) represents an infinitesimally small volume element within this region.
**Integral Explanation:**
The given integral computes the accumulation of the function \( 4x^3 + 3y^2 + 2x \) over the entire volume of the rectangular prism \( B \). This involves integrating with respect to \( z \), \( y \), and \( x \), typically in that order, although the order can vary based on the problem requirements.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd20dfe5a-a4c1-4793-9f05-a80ad59a67d4%2F3d7b7af5-69a7-45ed-874d-6d3d2412e843%2Fc4v8cod_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Triple Integral Evaluation over a Rectangular Prism**
**Problem Statement:**
Evaluate the triple integral:
\[
\iiint\limits_{B} (4x^3 + 3y^2 + 2x) \, dV
\]
where the region \( B \) is defined as:
\[
B = \{ (x, y, z) \mid 0 \leq x \leq 8, \, 0 \leq y \leq 8, \, 0 \leq z \leq 9 \}
\]
**Region Description:**
The region \( B \) is a rectangular prism (cuboid) in 3-dimensional space. Its dimensions are given by the intervals on \( x \), \( y \), and \( z \):
- \( x \) ranges from 0 to 8.
- \( y \) ranges from 0 to 8.
- \( z \) ranges from 0 to 9.
The volume \( dV \) represents an infinitesimally small volume element within this region.
**Integral Explanation:**
The given integral computes the accumulation of the function \( 4x^3 + 3y^2 + 2x \) over the entire volume of the rectangular prism \( B \). This involves integrating with respect to \( z \), \( y \), and \( x \), typically in that order, although the order can vary based on the problem requirements.
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