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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![**Problem Statement:**
Evaluate the iterated integral.
\[ \int_{0}^{\pi} \int_{0}^{2} \int_{0}^{\sqrt{4 - z^{2}}} z \sin(x) \, dy \, dz \, dx \]
**Explanation:**
This problem involves evaluating a triple integral of the function \( z \sin(x) \) over the given limits of integration.
**Steps for Evaluation:**
1. **Innermost Integral:**
Evaluate the innermost integral with respect to \( y \):
\[ \int_{0}^{\sqrt{4 - z^{2}}} z \sin(x) \, dy \]
2. **Middle Integral:**
Integrate the result of the first integral with respect to \( z \):
\[ \int_{0}^{2} \left( \int_{0}^{\sqrt{4 - z^{2}}} z \sin(x) \, dy \right) dz \]
3. **Outermost Integral:**
Finally, integrate the result of the second integral with respect to \( x \):
\[ \int_{0}^{\pi} \left( \int_{0}^{2} \left( \int_{0}^{\sqrt{4 - z^{2}}} z \sin(x) \, dy \right) dz \right) dx \]
**Diagrams and Graphs:**
There are no diagrams or graphs associated with this specific problem. The integral limits suggest a volume bounded by the specified ranges of \( x \), \( y \), and \( z \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe040b111-4b10-4c24-8337-c45ed461242d%2F74f41dee-30f9-4345-bc10-14a3ab538feb%2Fa2m3o2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Evaluate the iterated integral.
\[ \int_{0}^{\pi} \int_{0}^{2} \int_{0}^{\sqrt{4 - z^{2}}} z \sin(x) \, dy \, dz \, dx \]
**Explanation:**
This problem involves evaluating a triple integral of the function \( z \sin(x) \) over the given limits of integration.
**Steps for Evaluation:**
1. **Innermost Integral:**
Evaluate the innermost integral with respect to \( y \):
\[ \int_{0}^{\sqrt{4 - z^{2}}} z \sin(x) \, dy \]
2. **Middle Integral:**
Integrate the result of the first integral with respect to \( z \):
\[ \int_{0}^{2} \left( \int_{0}^{\sqrt{4 - z^{2}}} z \sin(x) \, dy \right) dz \]
3. **Outermost Integral:**
Finally, integrate the result of the second integral with respect to \( x \):
\[ \int_{0}^{\pi} \left( \int_{0}^{2} \left( \int_{0}^{\sqrt{4 - z^{2}}} z \sin(x) \, dy \right) dz \right) dx \]
**Diagrams and Graphs:**
There are no diagrams or graphs associated with this specific problem. The integral limits suggest a volume bounded by the specified ranges of \( x \), \( y \), and \( z \).
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