EXAMPLE 2 Evaluate SS F. dS, where F(x, Y, z) = 7xyi + (y² + e)j + cos(xy)k and S is the surface of the region E bounded by the parabolic cylinder z = 4 - x2 and the planes z = 0, y = 0, and y + z 6. (See the figure.)
EXAMPLE 2 Evaluate SS F. dS, where F(x, Y, z) = 7xyi + (y² + e)j + cos(xy)k and S is the surface of the region E bounded by the parabolic cylinder z = 4 - x2 and the planes z = 0, y = 0, and y + z 6. (See the figure.)
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 12T
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Question
![EXAMPLE 2 Evaluate (S F · dS, where
F(x, y, z) = 7xyi + (y2 + exz)j + cos(xy)k
and S is the surface of the region E bounded by the parabolic cylinder z = 4 - x2 and the planes z = 0, y = 0, and y + z = 6. (See
the figure.)
SOLUTION
It would be extremely difficult to evaluate the given surface integral directly. (We would have to evaluate four surface
integrals corresponding to the four pieces of S.) Furthermore, the divergence of F is much less complicated than F itself:
Video Example )
a
a
div F =
7xy
) +.
(y2 + exz) + -(cos(xy))
Əx
ду
az
7y + 2y
Therefore, we use the Divergence Theorem to transform the given surface integral into a triple integral. The easiest way to evaluate
the triple integral is to express E as a type 3 region:
E = {(x, y, z) | -2 s x s 2, 0 s z 4 - x2, 0 s y s 6 - z}
Then we have
SS F. dS = SSS div F dV = SSS 9y dV
4-z
y dy dz dx
- LL
dz dx
18-* dx
- 216]dx
+ 6x4 + 12x2 -
) dx](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11116739-cb46-40a0-8159-7a3629791d30%2Fb00bff0b-374b-4ea9-bfaa-8d871f8fc402%2Fx3ucla_processed.jpeg&w=3840&q=75)
Transcribed Image Text:EXAMPLE 2 Evaluate (S F · dS, where
F(x, y, z) = 7xyi + (y2 + exz)j + cos(xy)k
and S is the surface of the region E bounded by the parabolic cylinder z = 4 - x2 and the planes z = 0, y = 0, and y + z = 6. (See
the figure.)
SOLUTION
It would be extremely difficult to evaluate the given surface integral directly. (We would have to evaluate four surface
integrals corresponding to the four pieces of S.) Furthermore, the divergence of F is much less complicated than F itself:
Video Example )
a
a
div F =
7xy
) +.
(y2 + exz) + -(cos(xy))
Əx
ду
az
7y + 2y
Therefore, we use the Divergence Theorem to transform the given surface integral into a triple integral. The easiest way to evaluate
the triple integral is to express E as a type 3 region:
E = {(x, y, z) | -2 s x s 2, 0 s z 4 - x2, 0 s y s 6 - z}
Then we have
SS F. dS = SSS div F dV = SSS 9y dV
4-z
y dy dz dx
- LL
dz dx
18-* dx
- 216]dx
+ 6x4 + 12x2 -
) dx
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