Explanation Given Information: A region is bounded above by the paraboloid having an equation z = 4 − x 2 − y 2 and below by the plane having an equation z + 2 y = 4 Formula Used: To define the double integral of a continuous function f ( x , y ) over a non-rectangular domain D in the x y − plane , so enclose D within a rectangle say R where R = [ a , b ] × [ c , d ] Define a function F ( x , y ) on R as follows: F ( x , y ) = { f ( x , y ) ∀ ( x , y ) ∈ D 0 ∀ ( x , y ) ∈ D Now, the double Integral of f ( x , y ) over D is defined as the double integral of F and R that is ∬ D f ( x , y ) d A = ∬ R F ( x , y ) d A F ( x , y ) is continuous as it depends on f ( x , y ) which is continuous on D . Calculation: The volume of the region D is given by D = ∭ D d V where 4 − 2 y ≤ z ≤ 4 − x 2 − y 2 ⇒ ∬ A 2 y − x 2 − y 2 d A Where, A is the projection of D on the x y − plane , but in x y − plane z = 0 , therefore A is given by: 4 − x 2 − y 2 = 4 − 2 y ⇒ 2 y − x 2 − y 2 = 0 ⇒ x 2 + ( y − 1 ) 2 = 1 In the first quadrant for any point ( x , y ) ∈ A 0 ≤ y ≤ 2 , 0 ≤ x ≤ 2 y − y 2 hence volume of D is given by: D = ∫ 0 2 ∫ 0

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ISBN: 9781524916817

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

Publisher: Kendall Hunt Publishing

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Chapter 12, Problem 64SP

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To find: The volume of the region bounded by the given paraboloid.

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Chapter 12, Problem 63SP

Chapter 12, Problem 65SP

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The volume of the region bounded by the given paraboloid.

Solution Summary: The author explains how to define the double integral of a continuous function f(x,y) over D in the xy-plane.

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To find: The volume of the region bounded by the given paraboloid.

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Chapter 12 Solutions