Find the average value of F(x, y, z) over the given region. F(x, y, z) = x2 + 9 over the cube in the first octant bounded by the coordinate planes and the planes x = 2, y = 2, and z =2.

DofotheroU 2021-02-24 Answered

Find the average value of F(x, y, z) over the given region. F(x,y,z)=x2+9 over the cube in the first octant bounded by the coordinate planes and the planes x=2,y=2,and z=2.

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Expert Answer

Roosevelt Houghton
Answered 2021-02-25 Author has 106 answers

Let D be the volume of the cube bounded by the coordinate planes and the planes x=2,y=2,and z=2.
Then D=222=8
Calculate average value of x2+9 over give cube D using formula:
Average value =1volume of DDFdV
Volume of D is equal to 23=8
Average value =1volume of D020202(x2+9)dzdydx
=180202(x2z+9z)02dydx (Integrate in relation to z.)
=180202(2x2+18)dyxe (Compute the boundaries)
=1802(2x2y+18y)02dx (Intergrate in relation to y)
=1802(4x2+36)dx (Compute the boundaries)
=18(4x33+36x)02 (Intergrate in relation to x)
=18(323+72) (Compute the boundaries)
=43+9=313
Final answer =313
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