Find the triple integral R R RB (xyz) dV .

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the triple integral over the tridimensional region `B = [a,b]x[c,d]x[e,f]` , such that:

`int int_B int (xyz) dV = int_a^b int_c^d int_e_f (xyz)dz dx dy`

`int_a^b int_c^d xy*z^2/2|_e^f dx dy = int_a^b int_c^d xy (f^2 - e^2)/2 dx dy`

`int_a^b int_c^d xy (f^2 - e^2)/2 dx dy = (f^2 - e^2)/2 int_a^b y*x^2/2|_c^d dy`

`int_a^b int_c^d xy (f^2 - e^2)/2 dx dy = ((f^2 - e^2)(d^2 - c^2))/4 int_a^b ydy`

`int_a^b int_c^d xy (f^2 - e^2)/2 dx dy = ((f^2 - e^2)(d^2 - c^2))/4*y^2/2|_a^b`

`int_a^b int_c^d xy (f^2 - e^2)/2 dx dy = ((f^2 - e^2)(d^2 - c^2)(b^2-a^2))/8`

Hence, evaluating the triple integral, over the tridimensional region `B = [a,b]x[c,d]x[e,f]` , yields `int int_B int (xyz) dV = ((f^2 - e^2)(d^2 - c^2)(b^2-a^2))/8.`

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