Use cylindrical coordinates to evaluate the integral where D is the solid bounded above by the plane z = 2 and below by the surface 2z = x² + y². (see the figure on page 841) -1/2 [[[ z(x² + y²)-¹/² dx dy dz =
Use cylindrical coordinates to evaluate the integral where D is the solid bounded above by the plane z = 2 and below by the surface 2z = x² + y². (see the figure on page 841) -1/2 [[[ z(x² + y²)-¹/² dx dy dz =
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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Question
![**Problem Description:**
Use cylindrical coordinates to evaluate the integral where \( D \) is the solid bounded above by the plane \( z = 2 \) and below by the surface \( 2z = x^2 + y^2 \). (See the figure on page 841)
\[
\iiint\limits_{D} z(x^2 + y^2)^{-1/2} \, dx \, dy \, dz = \text{[provide result here]}
\]
**Explanation:**
In this problem, you are asked to evaluate a triple integral using cylindrical coordinates. The region \( D \) is defined by an upper plane \( z = 2 \) and a lower surface described by the equation \( 2z = x^2 + y^2 \).
**Steps to Solve:**
1. **Convert to Cylindrical Coordinates:**
- Recognize that \( x^2 + y^2 = r^2 \), where \( r \) is the radial distance in cylindrical coordinates.
- Substitute these values into the integral.
2. **Set Up the Bounds:**
- The surface \( 2z = r^2 \) transforms into the equation for lower \( z \)-bounds.
- The upper bound for \( z \) is the plane \( z = 2 \).
3. **Evaluate the Integral:**
- The integral setup involves integration over \( r \), \( \theta \), and \( z \), using appropriate bounds.
- Express \( z(x^2 + y^2)^{-1/2} \) as \( z(r^2)^{-1/2} \).
This exercise is intended to provide practice in transforming and evaluating integrals in cylindrical coordinates, while thoroughly considering geometric constraints.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39ee3059-6733-456c-bbc4-46bedd3c18f2%2Ffcc08b3a-a88e-487e-9ddb-712d8044c70c%2Fq926p3a_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Description:**
Use cylindrical coordinates to evaluate the integral where \( D \) is the solid bounded above by the plane \( z = 2 \) and below by the surface \( 2z = x^2 + y^2 \). (See the figure on page 841)
\[
\iiint\limits_{D} z(x^2 + y^2)^{-1/2} \, dx \, dy \, dz = \text{[provide result here]}
\]
**Explanation:**
In this problem, you are asked to evaluate a triple integral using cylindrical coordinates. The region \( D \) is defined by an upper plane \( z = 2 \) and a lower surface described by the equation \( 2z = x^2 + y^2 \).
**Steps to Solve:**
1. **Convert to Cylindrical Coordinates:**
- Recognize that \( x^2 + y^2 = r^2 \), where \( r \) is the radial distance in cylindrical coordinates.
- Substitute these values into the integral.
2. **Set Up the Bounds:**
- The surface \( 2z = r^2 \) transforms into the equation for lower \( z \)-bounds.
- The upper bound for \( z \) is the plane \( z = 2 \).
3. **Evaluate the Integral:**
- The integral setup involves integration over \( r \), \( \theta \), and \( z \), using appropriate bounds.
- Express \( z(x^2 + y^2)^{-1/2} \) as \( z(r^2)^{-1/2} \).
This exercise is intended to provide practice in transforming and evaluating integrals in cylindrical coordinates, while thoroughly considering geometric constraints.
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