Rewrite the following integral using the indicated order of integration and then evaluate the resulting integral. 7 0 4x+4 SS S 0-1 0 7 0 4x+4 SS S 0-1 0 dy dx dz in the order dz dx dy dy dx dz= 000 SS S 000 dz dx dy = (Simplify your answer.)
Rewrite the following integral using the indicated order of integration and then evaluate the resulting integral. 7 0 4x+4 SS S 0-1 0 7 0 4x+4 SS S 0-1 0 dy dx dz in the order dz dx dy dy dx dz= 000 SS S 000 dz dx dy = (Simplify your answer.)
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 Order of Integration
#### Problem Statement:
Rewrite the following integral using the indicated order of integration and then evaluate the resulting integral.
\[ \int_{0}^{7} \int_{-1}^{0} \int_{0}^{4x+4} \, dy \, dx \, dz \]
in the order \( dz \, dx \, dy \).
#### Solution:
1. **Change the Order of Integration:**
Given integral:
\[ \int_{0}^{7} \int_{-1}^{0} \int_{0}^{4x+4} \, dy \, dx \, dz \]
Rewritten in the order \( dz \, dx \, dy \):
\[ \int_{\text{(limits for dy)}} \int_{\text{(limits for dx)}} \int_{\text{(limits for dz)}} \, dz \, dx \, dy \]
To identify the appropriate limits for \( dz \, dx \, dy \), note the limits of the variables from the given order, analyze each integral's boundary:
The innermost integral with respect to \( dy \) runs from 0 to \( 4x + 4 \).
The next integral \( dx \) runs from -1 to 0.
The outermost integral \( dz \) runs from 0 to 7.
Substitute these limits appropriately.
2. **Evaluate the Integral:**
Compute the integral:
Evaluating \(\int_{0}^{7} \int_{-1}^{0} \int_{0}^{4x+4} dy \, dz \, dx \).
Since \( \int_{0}^{4x+4} dy = 4x + 4 \),
\[ \int_{0}^{7} \int_{-1}^{0} (4x + 4) \, dz \, dx \]
Integrate with respect to \( dz \),
\[ \int_{0}^{7} \int_{-1}^{0} 4(x + 1) \, dz \, dx \]
Integrate with respect to \( dy \),
Finally, simplify the result.
#### Final Step:
Simplify your answer in the solution box provided.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf012473-cbdd-4844-9f9c-11a895f96b45%2F43d607c4-dacb-4489-b041-b288b8834db8%2Fv8y5ete_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Triple Integral Order of Integration
#### Problem Statement:
Rewrite the following integral using the indicated order of integration and then evaluate the resulting integral.
\[ \int_{0}^{7} \int_{-1}^{0} \int_{0}^{4x+4} \, dy \, dx \, dz \]
in the order \( dz \, dx \, dy \).
#### Solution:
1. **Change the Order of Integration:**
Given integral:
\[ \int_{0}^{7} \int_{-1}^{0} \int_{0}^{4x+4} \, dy \, dx \, dz \]
Rewritten in the order \( dz \, dx \, dy \):
\[ \int_{\text{(limits for dy)}} \int_{\text{(limits for dx)}} \int_{\text{(limits for dz)}} \, dz \, dx \, dy \]
To identify the appropriate limits for \( dz \, dx \, dy \), note the limits of the variables from the given order, analyze each integral's boundary:
The innermost integral with respect to \( dy \) runs from 0 to \( 4x + 4 \).
The next integral \( dx \) runs from -1 to 0.
The outermost integral \( dz \) runs from 0 to 7.
Substitute these limits appropriately.
2. **Evaluate the Integral:**
Compute the integral:
Evaluating \(\int_{0}^{7} \int_{-1}^{0} \int_{0}^{4x+4} dy \, dz \, dx \).
Since \( \int_{0}^{4x+4} dy = 4x + 4 \),
\[ \int_{0}^{7} \int_{-1}^{0} (4x + 4) \, dz \, dx \]
Integrate with respect to \( dz \),
\[ \int_{0}^{7} \int_{-1}^{0} 4(x + 1) \, dz \, dx \]
Integrate with respect to \( dy \),
Finally, simplify the result.
#### Final Step:
Simplify your answer in the solution box provided.
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