Use the Divergence Theorem to calculate the surface integral F(x, y, z) = 3xy²i + xe²j + z³k, S is the surface of the solid bounded by the cylinder y2 + z² = 4. and the planes x = -2 and x = 4 -//L For F(x, y, z) = 3xy² + xe²j + z³k we have 3(y² +z²) 3y² +3:2 Step 1 If the surface S has positive orientation and bounds the simple solid E, then the Divergence Theorem tells us that 16 F F. ds = div F = SSF. Submit Step 2 Since S bounds the cylinder y2 + z² = 4 between the planes x = -2 and x = 4, we will use cylindrical coordinates, with polar coordinates in the yz-plane. Therefore, y = r cos(8), z = r sin(8), and x = x. We, therefore, have the following. F. ds = div F dV. SSS div F dV = 6²² 6³² L² (³r² (3r² cos² (0) 2π - 6³6LC [[₁ 37.3 dx dr de = 3 Skip (you cannot come back) F. ds; that is, calculate the flux of F across S. 3r² sin²(0) 37.3 Step 3 This triple integral can be broken into a product of integrals and evaluated, as follows. -2π 1²³ 1² / 2² (3²³) 3 [2h de 6²[ dr Ldx -2 3(2π)( (6) 3r² sin² (e)r dx dr de dx dr de
Use the Divergence Theorem to calculate the surface integral F(x, y, z) = 3xy²i + xe²j + z³k, S is the surface of the solid bounded by the cylinder y2 + z² = 4. and the planes x = -2 and x = 4 -//L For F(x, y, z) = 3xy² + xe²j + z³k we have 3(y² +z²) 3y² +3:2 Step 1 If the surface S has positive orientation and bounds the simple solid E, then the Divergence Theorem tells us that 16 F F. ds = div F = SSF. Submit Step 2 Since S bounds the cylinder y2 + z² = 4 between the planes x = -2 and x = 4, we will use cylindrical coordinates, with polar coordinates in the yz-plane. Therefore, y = r cos(8), z = r sin(8), and x = x. We, therefore, have the following. F. ds = div F dV. SSS div F dV = 6²² 6³² L² (³r² (3r² cos² (0) 2π - 6³6LC [[₁ 37.3 dx dr de = 3 Skip (you cannot come back) F. ds; that is, calculate the flux of F across S. 3r² sin²(0) 37.3 Step 3 This triple integral can be broken into a product of integrals and evaluated, as follows. -2π 1²³ 1² / 2² (3²³) 3 [2h de 6²[ dr Ldx -2 3(2π)( (6) 3r² sin² (e)r dx dr de dx dr de
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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![Use the Divergence Theorem to calculate the surface integral
F(x, y, z) = 3xy²i + xe²j + z³k,
S is the surface of the solid bounded by the cylinder y2 + z² = 4. and the planes x = -2 and x = 4
div F =
Step 1
If the surface S has positive orientation and bounds the simple solid E, then the Divergence Theorem tells us that
[ [₁ F · ds = [ ] [
For F(x, y, z) = 3xy²i + xe²j + z³k we have
3 (₁²+z²)
[[ / ] [
= S
F. ds =
div F dV.
=
Submit
Step 2
Since S bounds the cylinder y² + z² = 4 between the planes x = -2 and x = 4, we will use cylindrical
coordinates, with polar coordinates in the yz-plane. Therefore, y = r cos(8), z = r sin(0), and x = x. We,
therefore, have the following.
3y² +3:²
div F dV
2π
2
4
- [² TL²
=
.2π
2
4
[²*²*₁ (3r² cos² (0) + 3,² sin²(0)
37.3
2π
2
4
*[²L²₂ (3²³) dx dr do =
-2
=
Skip (you cannot come back)
Step 3
This triple integral can be broken into a product of integrals and evaluated, as follows.
3
SS. F
-2π
de
3(2π)(
2
F. ds; that is, calculate the flux of F across S.
10
37.3
(6)
3r² sin² (0)
dx dr de
dr
4
dx
dx dr de](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa80fd0a7-9ae8-4bc5-b926-d69bfc10c0d7%2F0c290269-73ff-44cc-bfe7-332f98acb5ef%2F76jub1e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Use the Divergence Theorem to calculate the surface integral
F(x, y, z) = 3xy²i + xe²j + z³k,
S is the surface of the solid bounded by the cylinder y2 + z² = 4. and the planes x = -2 and x = 4
div F =
Step 1
If the surface S has positive orientation and bounds the simple solid E, then the Divergence Theorem tells us that
[ [₁ F · ds = [ ] [
For F(x, y, z) = 3xy²i + xe²j + z³k we have
3 (₁²+z²)
[[ / ] [
= S
F. ds =
div F dV.
=
Submit
Step 2
Since S bounds the cylinder y² + z² = 4 between the planes x = -2 and x = 4, we will use cylindrical
coordinates, with polar coordinates in the yz-plane. Therefore, y = r cos(8), z = r sin(0), and x = x. We,
therefore, have the following.
3y² +3:²
div F dV
2π
2
4
- [² TL²
=
.2π
2
4
[²*²*₁ (3r² cos² (0) + 3,² sin²(0)
37.3
2π
2
4
*[²L²₂ (3²³) dx dr do =
-2
=
Skip (you cannot come back)
Step 3
This triple integral can be broken into a product of integrals and evaluated, as follows.
3
SS. F
-2π
de
3(2π)(
2
F. ds; that is, calculate the flux of F across S.
10
37.3
(6)
3r² sin² (0)
dx dr de
dr
4
dx
dx dr de
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