2 (2) Compute SSS 2 Sart (v²ty ²) du using Spherical Coordinates. E= {(x, y, z) | x ²³7 y ² + z ² ≤ 1₁ the first octant x>0₁y >o, z>0}, that is of the Unit ball. 2 (0) Simplifying x² + y² after Substitution of Spherical variables cracks it open, (ii) first Octant determine's limits on 0 and (
2 (2) Compute SSS 2 Sart (v²ty ²) du using Spherical Coordinates. E= {(x, y, z) | x ²³7 y ² + z ² ≤ 1₁ the first octant x>0₁y >o, z>0}, that is of the Unit ball. 2 (0) Simplifying x² + y² after Substitution of Spherical variables cracks it open, (ii) first Octant determine's limits on 0 and (
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter9: Multivariable Calculus
Section9.5: Double Integrals
Problem 4YT
Related questions
Question
![### Computing Integrals using Spherical Coordinates
#### Problem Statement
Compute the triple integral of the function \(2\sqrt{x^2 + y^2}\) over the region \(E\) using spherical coordinates. The region \(E\) is defined as:
\[ E = \{(x, y, z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 \leq 1, x \geq 0, y \geq 0, z \geq 0 \} \]
This describes the first octant of the unit sphere.
#### Simplification and Solution Steps
1. **Substituting Spherical Coordinates:**
The function and integral will be simplified by substituting spherical variables. In spherical coordinates, \( x = \rho \sin\phi \cos\theta \), \( y = \rho \sin\phi \sin\theta \), and \( z = \rho \cos\phi \).
2. **Evaluating Simplification:**
Using these substitutions will simplify \( x^2 + y^2 \) inside the integrand.
3. **Limits in the First Octant:**
Since we are in the first octant, the limits for the spherical coordinates \(\rho\), \(\phi\), and \(\theta\) need to be determined.
- The radial coordinate \(\rho\) ranges from 0 to 1.
- The polar angle \(\phi\) ranges from 0 to \(\pi/2\).
- The azimuthal angle \(\theta\) ranges from 0 to \(\pi/2\).
By following these outlined steps, you can compute the integral within the specified region using spherical coordinates.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53b47ed7-4497-4fb9-9d0f-c41504626a91%2Fe28392b5-df54-4c75-b7de-5eb5abe1da6e%2F6e12n2r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Computing Integrals using Spherical Coordinates
#### Problem Statement
Compute the triple integral of the function \(2\sqrt{x^2 + y^2}\) over the region \(E\) using spherical coordinates. The region \(E\) is defined as:
\[ E = \{(x, y, z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 \leq 1, x \geq 0, y \geq 0, z \geq 0 \} \]
This describes the first octant of the unit sphere.
#### Simplification and Solution Steps
1. **Substituting Spherical Coordinates:**
The function and integral will be simplified by substituting spherical variables. In spherical coordinates, \( x = \rho \sin\phi \cos\theta \), \( y = \rho \sin\phi \sin\theta \), and \( z = \rho \cos\phi \).
2. **Evaluating Simplification:**
Using these substitutions will simplify \( x^2 + y^2 \) inside the integrand.
3. **Limits in the First Octant:**
Since we are in the first octant, the limits for the spherical coordinates \(\rho\), \(\phi\), and \(\theta\) need to be determined.
- The radial coordinate \(\rho\) ranges from 0 to 1.
- The polar angle \(\phi\) ranges from 0 to \(\pi/2\).
- The azimuthal angle \(\theta\) ranges from 0 to \(\pi/2\).
By following these outlined steps, you can compute the integral within the specified region using spherical coordinates.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 4 images
Recommended textbooks for you
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage