Explanation Given: ∫ ∫ S ( x y + e z ) d S where S is the triangle with the vertices (0,0,3), (1,0,2) and (0,4,1)as shown in the Figure. Formula used: Surface integral over the graph is ∫ ∫ S f ( x , y , z ) d S = ∫ ∫ S f ( x , y , g ( x , y ) ) 1 + g x 2 + g y 2 d x d y , where z = g ( x , y ) Calculation: Let the vertices of the triangle are P = ( 0 , 0 , 3 ) , Q = ( 1 , 0 , 2 ) and R = ( 0 , 4 , 1 ) Therefore, P Q → = ( 0 , 0 , 3 ) − ( 1 , 0 , 2 ) = ( − 1 , 0 , 1 ) And P R → = ( 0 , 0 , 3 ) − ( 0 , 4 , 1 ) = ( 0 , − 4 , 2 ) The normal vector is obtained by the cross product of these two vectors as P Q → × P R → = | i j k − 1 0 1 0 − 4 2 | = 4 i + 2 j + 4 k Thus, n = 〈 4 , 2 , 4 〉 Since the plane passes through the point ( 0 , 0 , 3 ) , thus, the equation of the plane is: 4 ( x − 0 ) + 2 ( y − 0 ) + 4 ( z − 3 ) = 0 4 x + 2 y + 4 z − 12 = 0 2 x + y + 2 z − 6 = 0 z = − 2 x − y + 6 2 ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 17.4, Problem 29E

To determine

To calculate:

$\int \int_{S} (x y + e^{z}) d S$ , where $S$ is the triangle in the Figure with the vertices (0,0,3), (1,0,2) and (0,4,1).

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Chapter 17.4, Problem 28E

Chapter 17.4, Problem 30E

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Chapter 17 Solutions

Calculus - Standalone book

Ch. 17.1 - Prob. 1PQ Ch. 17.1 - Prob. 2PQ Ch. 17.1 - Prob. 3PQ Ch. 17.1 - Prob. 4PQ Ch. 17.1 - Prob. 1E Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E

Knowledge Booster

∫ ∫ S ( x y + e z ) d S , where S is the triangle in the Figure with the vertices (0,0,3), (1,0,2) and (0,4,1).

Solution Summary: The author explains the formula used to calculate S, where S is the triangle with the vertices in the Figure.

To calculate:

$\int \int_{S} (x y + e^{z}) d S$ , where $S$ is the triangle in the Figure with the vertices (0,0,3), (1,0,2) and (0,4,1).

Want to see the full answer?

Chapter 17 Solutions

∫ ∫ S ( x y + e z ) d S , where S is the triangle in the Figure with the vertices (0,0,3), (1,0,2) and (0,4,1).

Solution Summary: The author explains the formula used to calculate S, where S is the triangle with the vertices in the Figure.

To calculate: ∫∫S(xy+ez)dS, where S is the triangle in the Figure with the vertices (0,0,3), (1,0,2) and (0,4,1).

Want to see the full answer?

Chapter 17 Solutions

To calculate:

$\int \int_{S} (x y + e^{z}) d S$ , where $S$ is the triangle in the Figure with the vertices (0,0,3), (1,0,2) and (0,4,1).