Use cylindrical coordinates to evaluate the triple integral al S the circular paraboloid z = 9 -9 (x² + y²) and the xy -plane. x² + y² dV, where E is the solid bounded
Use cylindrical coordinates to evaluate the triple integral al S the circular paraboloid z = 9 -9 (x² + y²) and the xy -plane. x² + y² dV, where E is the solid bounded
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Problem Statement:**
Use cylindrical coordinates to evaluate the triple integral:
\[
\iiint_E \sqrt{x^2 + y^2} \, dV,
\]
where \( E \) is the solid bounded by the circular paraboloid \( z = 9 - 9(x^2 + y^2) \) and the \( xy \)-plane.
**Instructions:**
To solve this problem, follow these steps:
1. **Understand the region \( E \):**
- The region \( E \) is bounded by the circular paraboloid and the \( xy \)-plane. The paraboloid opens downward and intersects the \( xy \)-plane where \( z = 0 \).
2. **Convert to cylindrical coordinates:**
- Recall that \( x^2 + y^2 = r^2 \) in cylindrical coordinates, where \( x = r\cos\theta \), \( y = r\sin\theta \), and \( z = z \).
- The bounds for \( r \) are determined by setting \( z = 0: 9 = 9r^2 \Rightarrow r = 1 \).
- The bounds for \( \theta \) are from 0 to \( 2\pi \).
3. **Set up the integral:**
- The integrand \( \sqrt{x^2 + y^2} \) becomes \( r \), so the integral becomes:
\[
\int_0^{2\pi} \int_0^1 \int_0^{9 - 9r^2} r \, dz \, r \, dr \, d\theta
\]
4. **Evaluate the integral:**
- Integrate with respect to \( z \) first, then \( r \), and finally \( \theta \).
5. **Find the value of the integral:**
- Calculate the integral to find the volume of the solid \( E \) using the specified limits and substitutions.
This problem involves concepts of multivariable calculus including triple integrals and coordinate transformations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39ee3059-6733-456c-bbc4-46bedd3c18f2%2F1dec3f67-5478-4276-8d14-7722bf08d3b4%2Fh470ft3_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use cylindrical coordinates to evaluate the triple integral:
\[
\iiint_E \sqrt{x^2 + y^2} \, dV,
\]
where \( E \) is the solid bounded by the circular paraboloid \( z = 9 - 9(x^2 + y^2) \) and the \( xy \)-plane.
**Instructions:**
To solve this problem, follow these steps:
1. **Understand the region \( E \):**
- The region \( E \) is bounded by the circular paraboloid and the \( xy \)-plane. The paraboloid opens downward and intersects the \( xy \)-plane where \( z = 0 \).
2. **Convert to cylindrical coordinates:**
- Recall that \( x^2 + y^2 = r^2 \) in cylindrical coordinates, where \( x = r\cos\theta \), \( y = r\sin\theta \), and \( z = z \).
- The bounds for \( r \) are determined by setting \( z = 0: 9 = 9r^2 \Rightarrow r = 1 \).
- The bounds for \( \theta \) are from 0 to \( 2\pi \).
3. **Set up the integral:**
- The integrand \( \sqrt{x^2 + y^2} \) becomes \( r \), so the integral becomes:
\[
\int_0^{2\pi} \int_0^1 \int_0^{9 - 9r^2} r \, dz \, r \, dr \, d\theta
\]
4. **Evaluate the integral:**
- Integrate with respect to \( z \) first, then \( r \), and finally \( \theta \).
5. **Find the value of the integral:**
- Calculate the integral to find the volume of the solid \( E \) using the specified limits and substitutions.
This problem involves concepts of multivariable calculus including triple integrals and coordinate transformations.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
