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here you have the same question in the enclosed

What is the average value?

average value for (exp [tex]\alpha[/tex]Z ) = [tex]\int[/tex][tex]\int[/tex] exp[tex]\alpha[/tex]Z ds / [tex]\int[/tex][tex]\int[/tex] ds

Over the sphere S: X^2+Y^2+Z^2=a^2

Also by use the parameterization

X= a sinϴ cos ɸ

Y= a sinϴ sin ɸ

Z= a cos ϴ

And the usual substitution t = cos ϴ in the integral:

[tex]0\int[/tex][tex]\Pi[/tex] f (cos ϴ) sinϴ dϴ =[tex]-1\int[/tex]1 f(t) dt

Thanks

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