Write an integral for the average value of f(x, y,z)equals=xyz over the region bounded by the paraboloid xxequals=8181minus−yysquared2minus− zzsquared2 and the yyzz-plane (assuming the volume of the region is known).

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!!!

(sin 42)/31. = sinx)/37

sinx = 37 (sin42)/31

= 37× 0.66/31 = 0.787

x= 51.9

other angle opposite to y = 180-51.9-42

= 180- 93.9 =86.1

so

sin42)/31 = sin 86.1)/y

0.021 = 0.997 /y

y= 0.997/0.021 =47.47