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=
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)).
To evaluate the given triple integral, we follow these steps:
Step 1: Identify the limits of integration for each variable.
Given limits:
3 ≤ x ≤ 6
0 ≤ y ≤ 8
0 ≤ z ≤ 2
Step 2: Set up the triple integral.
∫∫∫Bf(x,y,z)dV = ∫∫∫b (z/x) dV
Step 3: Evaluate the triple integral.
∫∫∫b (z/x) dV = ∫∫∫ (z/x) dx dy dz
Integration with respect to x and y will be done first. Integration with respect to z will be done last.
∫∫∫ (z/x) dx dy dz = ∫∫ ∫ (z/x) [from x=3 to x=6] dy dz
Now perform integration with respect to x.
∫∫∫ (z/x) [from x=3 to x=6] dy dz = ∫∫ (z ln|x|) [from x=3 to x=6] dy dz
Next, perform integration with respect to y.
∫∫ (z ln|x|) [from x=3 to x=6] dy dz = ∫ (6z ln|x| - 3z ln|x|) [from y=0 to y=8] dz
Now integrate with respect to z.
∫ (6z ln|x| - 3z ln|x|) [from y=0 to y=8] dz = (6ln|x| - 3ln|x|) [from z=0 to z=2]
Finally, evaluate the limits.
= (6ln|6| - 3ln|6|) - (6ln|3| - 3ln|3|)
Simplifying further:
= 3ln(2) - 3ln(3)
Therefore, the value of the given triple integral is 3ln(2) - 3ln(3).