4. Let W be the solid region in the first octant that lies below the plane z = 6 and inside the cylinder 2² + y? = 4 (see figure to the right). Let F = zi+ (xy + 1) +(ye*) k. (a) Calculate fs, F dà where Si is the left surface of the solid region W (that is, the portion of W that lies in the plane y = 0), oriented in the positive y direction. (b) Calculate fs, F-dÃ, where S2 is the curved portion of W that lies on the cylinder, oriented away from the origin.
4. Let W be the solid region in the first octant that lies below the plane z = 6 and inside the cylinder 2² + y? = 4 (see figure to the right). Let F = zi+ (xy + 1) +(ye*) k. (a) Calculate fs, F dà where Si is the left surface of the solid region W (that is, the portion of W that lies in the plane y = 0), oriented in the positive y direction. (b) Calculate fs, F-dÃ, where S2 is the curved portion of W that lies on the cylinder, oriented away from the origin.
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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![**Vector Field and Surface Integrals in the First Octant**
4. Consider the solid region \( W \) located in the first octant, bounded below by the plane \( z = 6 \) and within the cylinder \( x^2 + y^2 = 4 \). Refer to the accompanying diagram showing these boundaries. The vector field is given by:
\[
\mathbf{F} = z \, \mathbf{i} + (xy + 1) \, \mathbf{j} + (ye^z) \, \mathbf{k}
\]
### Tasks:
(a) **Calculate the Surface Integral over \( S_1 \):**
Perform the integral \(\iint_{S_1} \mathbf{F} \cdot d\mathbf{A}\), where \( S_1 \) is defined as the left surface of the solid region \( W \), i.e., the section of \( W \) along the plane \( y = 0 \). This portion is oriented in the positive \( y \)-axis direction.
(b) **Calculate the Surface Integral over \( S_2 \):**
Compute the integral \(\iint_{S_2} \mathbf{F} \cdot d\mathbf{A}\), where \( S_2 \) is the curved surface of \( W \) that resides on the cylindrical boundary. This part is oriented away from the origin.
### Diagram Explanation:
The 3D diagram illustrates the described solid region in the first octant. It features:
- The cylindrical boundary represented by \( x^2 + y^2 = 4 \).
- The plane \( z = 6 \).
- The relevant planes and surfaces labeled with axes \( x \), \( y \), and \( z \).
This setup is used to evaluate the surface integrals for different components of the solid region encompassed within these geometric constraints.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6527fc37-ebd0-4042-a7b2-b6e43e072f32%2F04de7fce-b993-4d1f-a658-542281f92978%2Feq9z6ua_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Vector Field and Surface Integrals in the First Octant**
4. Consider the solid region \( W \) located in the first octant, bounded below by the plane \( z = 6 \) and within the cylinder \( x^2 + y^2 = 4 \). Refer to the accompanying diagram showing these boundaries. The vector field is given by:
\[
\mathbf{F} = z \, \mathbf{i} + (xy + 1) \, \mathbf{j} + (ye^z) \, \mathbf{k}
\]
### Tasks:
(a) **Calculate the Surface Integral over \( S_1 \):**
Perform the integral \(\iint_{S_1} \mathbf{F} \cdot d\mathbf{A}\), where \( S_1 \) is defined as the left surface of the solid region \( W \), i.e., the section of \( W \) along the plane \( y = 0 \). This portion is oriented in the positive \( y \)-axis direction.
(b) **Calculate the Surface Integral over \( S_2 \):**
Compute the integral \(\iint_{S_2} \mathbf{F} \cdot d\mathbf{A}\), where \( S_2 \) is the curved surface of \( W \) that resides on the cylindrical boundary. This part is oriented away from the origin.
### Diagram Explanation:
The 3D diagram illustrates the described solid region in the first octant. It features:
- The cylindrical boundary represented by \( x^2 + y^2 = 4 \).
- The plane \( z = 6 \).
- The relevant planes and surfaces labeled with axes \( x \), \( y \), and \( z \).
This setup is used to evaluate the surface integrals for different components of the solid region encompassed within these geometric constraints.
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