Find $ \ Bex I = \ iiint_v | x + y + 2Z | \ cdot | 4x + 4y-z | \ RD x \ RD Y \ RD Z, \ EEx $ where $ V $ is the region $ \ DPS {x ^ 2 + y ^ 2 + \ frac {z ^ 2} {4} \ Leq 1} $.
Q: Change $ \ Bex x = u, \ quad y = V, \ quad \ frac {z} {2} = W, \ EEx $ then $ \ beex \ Bea I & =\ iiint _ {u ^ 2 + V ^ 2 + w ^ 2 \ Leq 1} | u + V + 4 w | \ cdot | 4u + 4v-2w | \ cdot 2 \ rd u \ rd v \ rd w \\\& = 4 \ iiint _ {u ^ 2 + V ^ 2 + w ^ 2 \ leq 1} | u + V + 4 w | \ cdot | 2u + 2v-w | \ rd u \ rd v \ rd w. \ EEA \ eeex $ change $ \ Bex \ Tilde U =\frac {u + V + 4 w} {3 \ SQRT {2 }}, \ quad \ Tilde v =\frac {2u + 2v-w} {3}, \ quad \ Tilde W =\frac {-U + V }{\ SQRT {2 }}, \ EEx $ then $ \ beex \ Bea I & = 4 \ iiint _ {\ Tilde U ^ 2 + \ Tilde V ^ 2 + \ Tilde w ^ 2 \ Leq 1} | 3 \ SQRT {2} \ Tilde u | \ cdot | 3 \ Tilde v | \ RD \ Tilde U \ RD \ Tilde V \ RD \ Tilde w \ & = 36 \ SQRT {2} \ iiint _ {x ^ 2 + y ^ 2 + Z ^ 2 \ Leq 1} | XY | \ RD x \ rd y \ rd z \ & = 144 \ SQRT {2} \ iiint _ {x ^ 2 + y ^ 2 + Z ^ 2 \ Leq 1 \ atop x \ geq 0, Y \ geq 0} XY \ RD x \ rd y \ rd z \ & = 144 \ SQRT {2} \ int _ {-1} ^ 1 \ rd z \ iint _ {x ^ 2 + y ^ 2 \ Leq 1-z ^ 2 \ atop x \ geq0, Y \ geq 0} XY \ RD x \ rd y \ & = 144 \ SQRT {2} \ int _ {-1} ^ 1 \ rd z \ int_0 ^ {\ SQRT {1-z ^ 2 }}\ RD r \ int_0 ^ \ frac {\ PI} {2} r \ cos \ Theta \ cdot r \ sin \ Theta \ cdot r \ RD \ Theta \\<=\ frac {96 \ SQRT {2 }}{ 5 }. \ EEA \ eeex $
[Journal of mathematics at home lidu University] issue 241st using orthogonal transformation and symmetry to solve triple points