Write a triple integral, including limits of integration, that gives the volume between 3x +2y + z = 3 and 5x + 3y + z = 3 and above x+y≤ 1, x ≥ 0, y ≥ 0. volume = where a = C = e = b = d = and f = d d d
Write a triple integral, including limits of integration, that gives the volume between 3x +2y + z = 3 and 5x + 3y + z = 3 and above x+y≤ 1, x ≥ 0, y ≥ 0. volume = where a = C = e = b = d = and f = d d d
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Triple Integral for Volume Calculation**
*Instructions*: Write a triple integral, including limits of integration, that gives the volume between the planes \(3x + 2y + z = 3\) and \(5x + 3y + z = 3\), and is above the region defined by \(x + y \leq 1\), \(x \geq 0\), \(y \geq 0\).
### Volume Calculation
\[
\text{volume} = \int_{a}^{b} \int_{c}^{d} \int_{e}^{f} \, dz \, dy \, dx
\]
### Limits of Integration
- \(a =\)
- \(b =\)
- \(c =\)
- \(d =\)
- \(e =\)
- \(f =\)
These limits define the region where the triple integral will be evaluated to find the volume between the given planes. The conditions \(x + y \leq 1\), \(x \geq 0\), and \(y \geq 0\) describe a triangular region in the xy-plane which bounds the area for integration. The functions \(3x + 2y + z = 3\) and \(5x + 3y + z = 3\) represent the planes confining the volume in the z-direction.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ee46fbe-dc71-4a56-bc58-37c83c920656%2F22c4ca49-79f3-47d9-bc0b-aaeb1657eaba%2Fuxaen3i_processed.png&w=3840&q=75)
Transcribed Image Text:**Triple Integral for Volume Calculation**
*Instructions*: Write a triple integral, including limits of integration, that gives the volume between the planes \(3x + 2y + z = 3\) and \(5x + 3y + z = 3\), and is above the region defined by \(x + y \leq 1\), \(x \geq 0\), \(y \geq 0\).
### Volume Calculation
\[
\text{volume} = \int_{a}^{b} \int_{c}^{d} \int_{e}^{f} \, dz \, dy \, dx
\]
### Limits of Integration
- \(a =\)
- \(b =\)
- \(c =\)
- \(d =\)
- \(e =\)
- \(f =\)
These limits define the region where the triple integral will be evaluated to find the volume between the given planes. The conditions \(x + y \leq 1\), \(x \geq 0\), and \(y \geq 0\) describe a triangular region in the xy-plane which bounds the area for integration. The functions \(3x + 2y + z = 3\) and \(5x + 3y + z = 3\) represent the planes confining the volume in the z-direction.
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