To evaluate the triple integral, we integrate the function f(x, y, z) over the region B.
∫∫∫B f(x, y, z) dV = ∫∫∫B z*x^3 dV
First, we need to determine the limits of integration for each variable:
For x: x ranges from 3 to 6.
For y: y ranges from 0 to 8.
For z: z ranges from 0 to 2.
Now, we can evaluate the triple integral using these limits:
∫∫∫B z*x^3 dV = ∫∫∫B z*x^3 dz dy dx
The innermost integral will be taken with respect to z:
∫∫∫B z*x^3 dz dy dx = ∫∫ (0 to 2) [∫ (0 to 8) [∫ (3 to 6) (z*x^3) dx] dy] dz
Now, we integrate with respect to x:
∫∫ (0 to 2) [∫ (0 to 8) [∫ (3 to 6) (z*x^3) dx] dy] dz
= ∫∫ (0 to 2) [∫ (0 to 8) (z*(x^4/4)) | (3 to 6) dy] dz
= ∫∫ (0 to 2) [∫ (0 to 8) (z*(81/4)) dy] dz
= ∫∫ (0 to 2) (z*(81/4)*8) dz
= ∫ (0 to 2) (648/4)*z dz
= (648/4) * [z^2/2] (0 to 2)
= (648/4) * [(2^2/2) - (0^2/2)]
= (648/4) * (2)
= 648
Therefore, ∫∫∫B f(x, y, z) dV = 648.
Evaluate ∫∫∫Bf(x,y,z)dV
for the specified function f
and B: f(x,y,z)=zx3≤x≤6,0≤y≤8,0≤z≤2
∫∫∫Bf(x,y,z)dV=
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