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Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Find the triple
over the box B = [0, a] × [0, b] × [0, c]
![**Triple Integral Calculation**
**Problem Statement:**
Find the following triple integral:
\[
\iiint_{B} 2 + x^2 y^2 \, dV
\]
where \( B \) is defined as:
\[
B = [0,1] \times [0,2] \times [0,3]
\]
**Explanation:**
This problem involves calculating a triple integral over the set \( B \), which is a rectangular prism—or a box—in three-dimensional space. The limits of integration are given by the intervals:
- \( x \) ranges from 0 to 1,
- \( y \) ranges from 0 to 2,
- \( z \) ranges from 0 to 3.
The integrand is \( 2 + x^2 y^2 \), a function of two of the variables, \( x \) and \( y \), which will be integrated with respect to the volume element \( dV \). The task involves integrating over each dimension sequentially, considering the given range for each variable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6077a9bd-c1d3-408f-b3b9-46142cdfb38a%2F4bdfaaa1-71ef-45f3-8f63-bc56526d46fb%2Fzbem0xb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Triple Integral Calculation**
**Problem Statement:**
Find the following triple integral:
\[
\iiint_{B} 2 + x^2 y^2 \, dV
\]
where \( B \) is defined as:
\[
B = [0,1] \times [0,2] \times [0,3]
\]
**Explanation:**
This problem involves calculating a triple integral over the set \( B \), which is a rectangular prism—or a box—in three-dimensional space. The limits of integration are given by the intervals:
- \( x \) ranges from 0 to 1,
- \( y \) ranges from 0 to 2,
- \( z \) ranges from 0 to 3.
The integrand is \( 2 + x^2 y^2 \), a function of two of the variables, \( x \) and \( y \), which will be integrated with respect to the volume element \( dV \). The task involves integrating over each dimension sequentially, considering the given range for each variable.
Expert Solution
Step 1
To find the triple integral , where B is the box .
dV=dzdydx.
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