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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![**Image Transcription and Explanation for Educational Website:**
Title: Converting Rectangular Integrals to Cylindrical Coordinates
---
**Text:**
Convert the integral
\[
\int_{0}^{5} \int_{0}^{y} \int_{0}^{1} (xy + z) \, dz \, dx \, dy
\]
into an integral in cylindrical coordinates.
**Solution in Cylindrical Coordinates:**
\[
\int_{\pi/4}^{\pi} \int_{0}^{\text{csc}(\theta)} \int_{0}^{0} \left( \right) \, dz \, dr \, d\theta
\]
---
**Explanation:**
The problem involves converting a given triple integral from rectangular (Cartesian) coordinates to cylindrical coordinates. Each of the variables \(x\), \(y\), and \(z\) will be transformed using the relationships:
- \(x = r \cos(\theta)\)
- \(y = r \sin(\theta)\)
- \(z = z\)
In cylindrical coordinates, \(\theta\) ranges from \(\pi/4\) to \(\pi\), and \(r\) ranges from 0 to \(\text{csc}(\theta)\). The variable \(z\) has a constant range, from 0 to itself (typically the upper limit derived from context, though in the diagram it appears cut-off). The function inside the integral will be expressed in terms of \(r\), \(\theta\), and \(z\).
**Key Points:**
1. **Integration Limits:**
- The integration limits adapt to the cylindrical representation, involving trigonometric functions like \(\text{csc}(\theta)\).
2. **Variable Transformation:**
- Cartesian \(x\)- and \(y\)-coordinates are replaced with cylindrical coordinates \(r\) and \(\theta\).
3. **Jacobian Determinant:**
- A common step in these conversions is adjusting for the Jacobian determinant which in cylindrical coordinates results in an additional \(r\) factor (though not visible here, it's crucial for complete transformation).
**Conclusion:**
The given expression illustrates the complexity of transforming integrals into different coordinate systems to simplify computation, especially in applications involving symmetry better suited for cylindrical or spherical coordinates.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e824278-0cf8-4697-a6ac-3768c642cb77%2F42f466b1-d89d-4958-a934-b58f5261f4d6%2Fk85kbo_processed.png&w=3840&q=75)
Transcribed Image Text:**Image Transcription and Explanation for Educational Website:**
Title: Converting Rectangular Integrals to Cylindrical Coordinates
---
**Text:**
Convert the integral
\[
\int_{0}^{5} \int_{0}^{y} \int_{0}^{1} (xy + z) \, dz \, dx \, dy
\]
into an integral in cylindrical coordinates.
**Solution in Cylindrical Coordinates:**
\[
\int_{\pi/4}^{\pi} \int_{0}^{\text{csc}(\theta)} \int_{0}^{0} \left( \right) \, dz \, dr \, d\theta
\]
---
**Explanation:**
The problem involves converting a given triple integral from rectangular (Cartesian) coordinates to cylindrical coordinates. Each of the variables \(x\), \(y\), and \(z\) will be transformed using the relationships:
- \(x = r \cos(\theta)\)
- \(y = r \sin(\theta)\)
- \(z = z\)
In cylindrical coordinates, \(\theta\) ranges from \(\pi/4\) to \(\pi\), and \(r\) ranges from 0 to \(\text{csc}(\theta)\). The variable \(z\) has a constant range, from 0 to itself (typically the upper limit derived from context, though in the diagram it appears cut-off). The function inside the integral will be expressed in terms of \(r\), \(\theta\), and \(z\).
**Key Points:**
1. **Integration Limits:**
- The integration limits adapt to the cylindrical representation, involving trigonometric functions like \(\text{csc}(\theta)\).
2. **Variable Transformation:**
- Cartesian \(x\)- and \(y\)-coordinates are replaced with cylindrical coordinates \(r\) and \(\theta\).
3. **Jacobian Determinant:**
- A common step in these conversions is adjusting for the Jacobian determinant which in cylindrical coordinates results in an additional \(r\) factor (though not visible here, it's crucial for complete transformation).
**Conclusion:**
The given expression illustrates the complexity of transforming integrals into different coordinate systems to simplify computation, especially in applications involving symmetry better suited for cylindrical or spherical coordinates.
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