Answer:
Average value = [(81 - y² - z²)⁴/96V
where V is the known volume.
Step-by-step explanation:
We're to find the average value of
f(x,y,z) = xyz
over the region bounded by paraboloid
x = (81 - y² - z²)
and the yz plane assuming the volume is known.
The average value = (the triple integral)/(volume)
The triple integral will be evaluated between the paraboloid and the origin and the yz plane and the origin.
Note that the differential volume dV used to evaluate the triple integral is given by
dV = dydzdx
The evaluation is presented in the attached image to this question.
After evaluating the triple integral, we obtain
Triple integral = [(81 - y² - z²)⁴/96]
If the known volume is V
Average value = [(81 - y² - z²)⁴/96V]
Hope this Helps!!!
