To evaluate the triple integral ∫∫∫Bf(x,y,z)dV, we need to compute the integral of f(x,y,z) over the region B defined by the given limits.
First, let's write the integral in its appropriate form:
∫∫∫Bf(x,y,z)dV = ∫∫∫B (z/x)dV
To evaluate this integral, we will use the order of integration dz dy dx.
Given that x ranges from 3 to 6, y ranges from 0 to 8, and z ranges from 0 to 2, we can compute the integral as follows:
∫∫∫B (z/x)dV = ∫[0,2] ∫[0,8] ∫[3,6] (z/x) dx dy dz
∫[0,2] ∫[0,8] ∫[3,6] (z/x) dx dy dz
= ∫[0,2] ∫[0,8] [ln(x)]|[3,6] dy dz (using the antiderivative of z/x, which is ln(x))
= ∫[0,2] ∫[0,8] (ln(6)-ln(3)) dy dz
= ∫[0,2] (ln(6)-ln(3))(y)|[0,8] dz
= ∫[0,2] (ln(6)-ln(3))(8-0) dz
= (ln(6)-ln(3))(8)(z)|[0,2]
= (ln(6)-ln(3))(8)(2-0)
= 16(ln(6)-ln(3))
Therefore, ∫∫∫Bf(x,y,z)dV = 16(ln(6)-ln(3)).
Evaluate ∫∫∫Bf(x,y,z)dV
for the specified function f and B:f(x,y,z)=z/x 3≤x≤6,0≤y≤8,0≤z≤2
∫∫∫Bf(x,y,z)dV=
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