Average value over a multivariable function using triple integrals. Find the average value of

Falak Kinney 2021-01-13 Answered

Average value over a multivariable function using triple integrals. Find the average value of F(x,y,z)=x2+y2+z2 over the cube in the first octant bounded bt the coordinate planes and the planes x=5, y=5, and z=5

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Expert Answer

Arnold Odonnell
Answered 2021-01-14 Author has 109 answers

Average value of F=1vEF(x,y,z)dV
Where, dV=dxdydzandV be the volume over the given region.
Now, x=[0,5], y=[0,5],z=[0,5]
Then find the Volume
V=z=05y=05x=05dxdydz
V=(z=05dx)(y=05dy)(x=05dz)
V=(x)05(y)05(z)05
V=5×5×5
V=125
Now Average value of F=1VEF(x,y,z)dV
=1125z=05y=05x=05(x2+y2+z2)dxdydz
=1125z=05y=05(x33+y2x+z2x)x=05dydz
=1125z=05y=05(1253+5y2+5z2)dydz
=1125z=05(1253y+5y33+5z2y)x=05
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