Miscellaneous surface integrals Evaluate the following integrals using the method of your choice. Assume normal vectors point either outward or upward . 53. ∬ S ( x , 0 , z ) x 2 + z 2 ⋅ n d S , where S is the cylinder x 2 + z 2 = a 2 , | y | ≤ 2
Miscellaneous surface integrals Evaluate the following integrals using the method of your choice. Assume normal vectors point either outward or upward . 53. ∬ S ( x , 0 , z ) x 2 + z 2 ⋅ n d S , where S is the cylinder x 2 + z 2 = a 2 , | y | ≤ 2
Miscellaneous surface integralsEvaluate the following integrals using the method of your choice. Assume normal vectors point either outward or upward.
53.
∬
S
(
x
,
0
,
z
)
x
2
+
z
2
⋅
n
d
S
, where S is the cylinder x2 + z2 = a2,
|
y
|
≤
2
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.
Evaluate the surface integral.
J y ds, S is the helicoid with vector equation r(u, v) = (u cos(v), u sin(v), v), 0sus 6,0 SV SR.
[(10) ()-1]
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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
Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise.
2x + 3y dx + e
-3y
dy, where C is the triangle with vertices (0, 0), (1, 0), (1, 1).
Thomas' Calculus: Early Transcendentals (14th Edition)
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