Explanation 1) Concept: S bea surface with equation z = g x , y then surface integral is ∬ S □ f ( x , y , z ) d S = ∬ D □ f ( r u , v ) r u × r v d A Where tangent vectors r u & r v defined by, r u = ∂ x ∂ u i + ∂ y ∂ u j + ∂ z ∂ u k & r v = ∂ x ∂ v i + ∂ y ∂ v j + ∂ z ∂ v k And D is the parameter domain. 2) Given: S is the part of plane z = 4 + x + y that lies inside the cylinder x 2 + y 2 = 4 3) Calculations: Let, f x , y , z = x 2 z + y 2 z The parametric surface S is the part of plane z = 4 + x + y that lies inside the cylinder x 2 + y 2 = 4 That is z = 4 + x + y with 0 ≤ x 2 + y 2 ≤ 4 is (by using x & y as parameters) r y , z = x i + y j + 4 + x + y k , x 2 + y 2 ≤ 4 Therefore, f r y , z = x 2 4 + x + y + y 2 4 + x + y = 4 + x + y x 2 + y 2 To find the tangent vectors r x a n d r y r x = i + o j + k r y = 0 i + j + k The normal vector to the tangent plane is r y × r z r x × r y = i j k 1 0 1 0 1 1 r x × r y = - i - j + k r x × r y = 1 + 1 + 1 r x × r y = 3 By using concept, the surface integral is, ∬ S □ f ( x , y , z ) d S = ∬ D □ f ( r x , y r x × r y d A The domain is D :

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ISBN: 9781285740621

Author: James Stewart

Publisher: Cengage Learning

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Chapter 16.R, Problem 28E

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The given surface integral

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Chapter 16.R, Problem 27E

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Chapter 16 Solutions

Calculus (MindTap Course List)

Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - Prob. 7E Ch. 16.1 - Prob. 8E Ch. 16.1 - Prob. 9E Ch. 16.1 - Prob. 10E

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The given surface integral

Solution Summary: The author explains that the parametric surface S is the part of plane z=4+x+y that lies inside the cylinder.

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To evaluate: The given surface integral

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Chapter 16 Solutions

To evaluate:

The given surface integral