Explanation Given: The paraboloid z = 3 x 2 + 3 y 2 and the cylinder x 2 + y 2 = 4 . Concept Used: The surface area of the surface given by z = f ( x , y ) over a region Dis given by: S = ∬ D ( f x ) 2 + ( f y ) 2 + 1 d A Calculation: To find the surface area of the given paraboloid z = 3 x 2 + 3 y 2 that lies inside the cylinder x 2 + y 2 = 4 , that is inside the disk of radius 2 centered at the origin, first find the partial derivatives of the given surface. To find the partial derivative of the function, first find the derivative of above function with respect to x , treating y as a constant, it gives f x = ∂ ∂ x ( 3 x 2 + 3 y 2 ) = 6 x Again, differentiating the given plane with respect to y , treating x as a constant, it gives f y = ∂ ∂ y ( 3 x 2 + 3 y 2 ) = 6 y Since the region is the disk of radius 2 centered at origin, so use the polar coordinates to evaluate the integral. The limit of integration in polar coordinates given by 0 ≤ r ≤ 2 , 0 ≤ θ ≤ 2 π So, the surface area of the paraboloid inside the given cylinder is S = ∬ D ( f x ) 2 + ( f y ) 2 + 1 d A = ∬ D ( 6 x ) 2 + ( 6 y ) 2 + 1 d A = ∬ D 36 x 2 + 36 y 2 + 1 d A = ∬ D 36 ( x 2 + y 2 ) + 1 d A = ∫ 0 2 π ∫ 0 2 36 ( r 2 ) + 1 r </

Student's Solution and Survival Manual for Calculus

7th Edition

ISBN: 9781524934040

Author: STRAUSS MONTY J, TODA MAGDALENA DANIELE, SMITH KARL J

Publisher: Kendall Hunt Publishing

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Chapter 12.4, Problem 10PS

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The surface area of the portion of the paraboloid z=3x2+3y2 that lies inside the cylinder x2+y2=4 .

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Chapter 12.4, Problem 9PS

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Student's Solution and Survival Manual for Calculus

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The surface area of the portion of the paraboloid z = 3 x 2 + 3 y 2 that lies inside the cylinder x 2 + y 2 = 4 .

Solution Summary: The author calculates the surface area of the paraboloid z = 3x2+3y2, which lies inside the cylinder.

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The surface area of the portion of the paraboloid z=3x2+3y2 that lies inside the cylinder x2+y2=4 .

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