Explanation Given: The equation of the part of the surface is given as follows. z = 1 + 3 x + 2 y 2 The required surface lies above the triangle with vertices ( 0 , 0 ) , ( 0 , 1 ) and ( 2 , 1 ) . Formula used: Write the expression to find the surface area of the plane. A ( S ) = ∬ D 1 + ( ∂ z ∂ x ) 2 + ( ∂ z ∂ y ) 2 d A (1) Calculation: Write the equation of part of the surface as follows. z = 1 + 3 x + 2 y 2 (2) Calculation of ∂ z ∂ x : Take partial derivative for equation (2) with respect to x . ∂ z ∂ x = ∂ ∂ x ( 1 + 3 x + 2 y 2 ) = 0 + 3 + 0 = 3 Calculation of ∂ z ∂ y : Take partial derivative for equation (2) with respect to y . ∂ z ∂ y = ∂ ∂ y ( 1 + 3 x + 2 y 2 ) = 0 + 0 + 4 y = 4 y Surface area of plane as follows: Substitute 3 for ∂ z ∂ x and 4 y for ∂ z ∂ y in equation (1), A ( S ) = ∬ D 1 + ( 3 ) 2 + ( 4 y ) 2 d A = ∬ D 1 + 9 + 16 y 2 d A = ∬ D 16 y 2 + 10 d A Simplify the expression as follows. A ( S ) = ∬ D 16 y 2 + 10 d A (3) As the required part of the surface lies above the triangle with vertices ( 0 , 0 ) , ( 0 , 1 ) and ( 2 , 1 ) , the region is inside the triangle. Now, obtain the equation of the line between ( 0 , 0 ) and ( 2 , 1 ) as follows. y − 0 x − 0 = 1 − 0 2 − 0 y x = 1 2 y = x 2 x = 2 y The region D is a triangle area with the x limit as 0 to 2 y and the limit y is 0 to 1. Therefore, the limit of x and y are written as follows

Essential Calculus: Early Transcendentals

2nd Edition

ISBN: 9781133112280

Author: James Stewart

Publisher: Cengage Learning

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Chapter 13.6, Problem 38E

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The area of the part of the surface z=1+3x+2y2 that lies above the triangle with vertices (0, 0), (0,1) and (2,1).

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Chapter 13.6, Problem 37E

Chapter 13.6, Problem 39E

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Chapter 13 Solutions

Essential Calculus: Early Transcendentals

Ch. 13.1 - Sketch the vector field F by drawing a diagram...Ch. 13.1 - Sketch the vector field F by drawing a diagram...Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Sketch the vector field F by drawing a diagram...Ch. 13.1 - Prob. 9E Ch. 13.1 - Sketch the vector field F by drawing a diagram...

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The area of the part of the surface z = 1 + 3 x + 2 y 2 that lies above the triangle with vertices ( 0 , 0 ) , ( 0 , 1 ) and ( 2 , 1 ) .

Solution Summary: The formula used to find the surface area of the plane is given as follows.

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The area of the part of the surface z=1+3x+2y2 that lies above the triangle with vertices (0, 0), (0,1) and (2,1).

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Chapter 13 Solutions