Surface integrals Evaluate the following surface integrals. 50. ∬ S 〈 0 , y , z 〉 ⋅ n d S ; S is the curved surface of the cylinder y 2 + z 2 = a 2 , | x | ≤ 8 with outward normal vectors.
Surface integrals Evaluate the following surface integrals. 50. ∬ S 〈 0 , y , z 〉 ⋅ n d S ; S is the curved surface of the cylinder y 2 + z 2 = a 2 , | x | ≤ 8 with outward normal vectors.
Solution Summary: The author evaluates the surface integral of a cylinder.
Surface integralsEvaluate the following surface integrals.
50.
∬
S
〈
0
,
y
,
z
〉
⋅
n
d
S
;
S is the curved surface of the cylinder
y
2
+
z
2
=
a
2
,
|
x
|
≤
8
with outward normal vectors.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Calculate ff f(x, y, z) d.S for the given surface and function.
x² + y² = 25, 0≤ z ≤ 4; f(x, y, z) = e¯²
Consider the shown work.
To =
T, =
аф
де
=
д
(5 cos 0, 5 sin 0, z) = (-5 sin 0, 5 cos 0, 0)
do
d
-(5 cos 0, 5 sin 0, z) = (0,0,1)
дz
i
N(0, z) = T₁ × T₂ = -5 sin 0
0
||N(0, z)|| =
5 cos 0
0
2π 4
[[ f(x, y, 2) ds = [²* ["^ e
S
(5 cos 0)² + (5 sin 0)² + 0 =
e² do dz
k
0 = (5 cos 0)i + (5 sin 0)j =
1
Identify the first error in the work shown.
/25 (cos² 0 + sin²0)
The surface integral is written incorrectly.
No errors exist in the work shown.
The parametrization of the cylinder is incorrect.
The normal vector N(0, z) is incorrect.
(5 cos 0, 5 sin 0, 0)
√25 = 5
Which of the following best describes the surface with the vector equation
r(u, v) = (cos u)i + (sin u)j + vk?
%3D
O A plane passing through the origin
O A cylinder whose axis is the z-axis
O A sphere of radius 1 centered at the origin
O A right circular cone whose vertex is the origin
Elementary Statistics: Picturing the World (7th Edition)
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