Explanation Given: The portion of the sphere x 2 + y 2 + z 2 = 8 that is inside the cone x 2 + y 2 − z 2 = 0 Calculation: For a continuous function f ( x , y ) with partial derivatives f x , f y in a given region R , the portion of the surface z = f ( x , y ) that lies over R has surface area S given as follows: S = ∫ ∫ R [ f x ( x , y ) ] 2 + 1 d A The objective is to find the surface area of the portion of the sphere x 2 + y 2 + z 2 = 8 that is inside the cone x 2 + y 2 − z 2 = 0 . First, the equation of the surface z = f ( x , y ) is required. x 2 + y 2 + z 2 = 8 z 2 = 8 − x 2 − y 2 z = 8 − x 2 − y 2 Hence f ( x , y ) = 8 − x 2 − y 2 Now for the region on x y -plane eliminate z as follows: x 2 + y 2 + z 2 = 8 x 2 + y 2 − z 2 = 0 2 ( x 2 + y 2 ) = 8 x 2 + y 2 = 4 Let x = r cos θ , y = r sin θ , thus x 2 + y 2 = 4 r 2 cos 2 θ + r 2 sin 2 θ = 4 r 2 = 4 r = 2 The boundary points are given by: 0 ≤ r ≤ 2 0 ≤ θ ≤ 2 π Also, f x ( x , y ) = ( − x ) 8 − x 2 − y 2 , f y ( x , y ) = ( − y ) 8 − x 2 − y 2 Thus, in Cartesian coordinates: [ f x ( x , y ) ] 2 + [ f y ( x , y ) ] 2 + 1 = [ ( − x ) 8 − x 2 − y 2 ] 2 + [ ( − y ) 8 − x 2 − y 2

6th Edition

ISBN: 9781465208880

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

Publisher: Kendall Hunt Publishing

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Chapter 12.4, Problem 21PS

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To find:The surface area of each surface.

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The surface area of each surface.

Solution Summary: The author explains that the objective is to find the surface area of a portion of the sphere z=f(x,y) that lies over R

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