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Jaydan Aguirre 2022-07-12 Answered
Compite the integral
V 4 z 2 ( x 2 + y 2 ) 3 d x d y d z ,
where V is the solid enclosed by the paraboloids x 2 + y 2 = z , x 2 + y 2 = 2 z and cones x 2 + y 2 = ( z 2 ) 2 , x 2 + y 2 = 4 ( z 2 ) 2 .
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Answers (1)

Antonio Dickerson
Answered 2022-07-13 Author has 5 answers
Let D = { ( r , z ) : r 0 , z 2 , z r 2 2 z , ( z 2 ) 2 r 2 4 ( z 2 ) 2 }, and consider the following change of variables: u = z / r 2 and v = ( z 2 ) 2 / r 2 , then
| det ( u r u z v r v z ) | = 2 4 z 2 r 5 .
Then
V { z 2 } 4 z 2 ( x 2 + y 2 ) 3 d x d y d z = 2 π D 4 z 2 r 6 r d r d z = 2 π u = 1 / 2 1 v = 1 / 4 1 1 2 d u d v = 3 π 8 .
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