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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![### Evaluating a Triple Integral by Hand
In this exercise, we are required to evaluate the following triple integral by hand:
\[ \int_{z=1}^{9} \int_{y=2}^{4} \int_{x=-1}^{2} \frac{x^4 \sqrt{z}}{y^3} \, dx \, dy \, dz \]
#### Steps to Solve:
1. **Set up the Integral:**
- The integral is set up in the standard order of integration from inner to outer: \(dx\), \(dy\), \(dz\).
2. **Integrate with respect to \(x\):**
- Evaluate the innermost integral (with respect to \(x\)) while treating \(y\) and \(z\) as constants:
\[
\int_{x=-1}^{2} \frac{x^4 \sqrt{z}}{y^3} \, dx
\]
3. **Integrate with respect to \(y\):**
- After simplifying the result from \(x\) integration, integrate with respect to \(y\):
\[
\int_{y=2}^{4} \left( \text{Result from the } x \text{-integration} \right) \, dy
\]
4. **Integrate with respect to \(z\):**
- Finally, integrate with respect to \(z\) using the result from the \(y\) integration:
\[
\int_{z=1}^{9} \left( \text{Result from the } y \text{-integration} \right) \, dz
\]
By following these steps in sequence, one can evaluate the given triple integral manually.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feffdcb57-6bec-4159-9775-276851093f6a%2Fa4e74709-c03a-43ca-b931-6590411815d6%2Fgz73did_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Evaluating a Triple Integral by Hand
In this exercise, we are required to evaluate the following triple integral by hand:
\[ \int_{z=1}^{9} \int_{y=2}^{4} \int_{x=-1}^{2} \frac{x^4 \sqrt{z}}{y^3} \, dx \, dy \, dz \]
#### Steps to Solve:
1. **Set up the Integral:**
- The integral is set up in the standard order of integration from inner to outer: \(dx\), \(dy\), \(dz\).
2. **Integrate with respect to \(x\):**
- Evaluate the innermost integral (with respect to \(x\)) while treating \(y\) and \(z\) as constants:
\[
\int_{x=-1}^{2} \frac{x^4 \sqrt{z}}{y^3} \, dx
\]
3. **Integrate with respect to \(y\):**
- After simplifying the result from \(x\) integration, integrate with respect to \(y\):
\[
\int_{y=2}^{4} \left( \text{Result from the } x \text{-integration} \right) \, dy
\]
4. **Integrate with respect to \(z\):**
- Finally, integrate with respect to \(z\) using the result from the \(y\) integration:
\[
\int_{z=1}^{9} \left( \text{Result from the } y \text{-integration} \right) \, dz
\]
By following these steps in sequence, one can evaluate the given triple integral manually.
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