Surface integrals using an explicit description Evaluate the surface integral ∬ S f ( x , y , z ) d S using an explicit representation of the surface . 38. f ( x , y , z ) = e z ; S is the plane z = 8 – x – 2 y in the first octant.
Surface integrals using an explicit description Evaluate the surface integral ∬ S f ( x , y , z ) d S using an explicit representation of the surface . 38. f ( x , y , z ) = e z ; S is the plane z = 8 – x – 2 y in the first octant.
Solution Summary: The author explains the surface integral displaystyleundersetSiintf(x,y,z)dS with the help of explicit description.
Surface integrals using an explicit descriptionEvaluate the surface integral
∬
S
f
(
x
,
y
,
z
)
d
S
using an explicit representation of the surface.
38.
f
(
x
,
y
,
z
)
=
e
z
; S is the plane z = 8 – x – 2y in the first octant.
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.
Using Stokes’ Theorem to evaluate a surface integral Evaluate∫∫S (∇ x F) # n dS, where F = -y i + x j + z k, in the following cases.a. S is the part of the paraboloid z = 4 - x2 - 3y2 that lies within the paraboloid z = 3x2 + y2 (the blue surface as shown). Assume n pointsin the upward direction on S.b. S is the part of the paraboloid z = 3x2 + y2 that lies within the paraboloidz = 4 - x2 - 3y2, with n pointing in the upward direction on S.c. S is the surface in part (b), but n pointing in the downward direction on S.
Surface integrals using an explicit description Evaluate the surface integral ∫∫S ƒ(x, y, z) dS using an explicit representation of the surface.
ƒ(x, y, z) = x2 + y2; S is the paraboloid z = x2 + y2, for 0 ≤ z ≤ 1.
Evaluate the surface integral F*n d-sigma where
F = -4xyi + 10x^2j - 4xyzk and S is the surface z = xe^y, x is between 0 and 1, y is between 0 and 1, with upwards orientation.
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