We start with:
#Z_S= int_0^hint_0^(2pi)int_0^(R(1-z/h)) mu_0 z rho^2 drho dvarphidz#
We can pull out the #mu_0#:
#= mu_0 int_0^hint_0^(2pi)int_0^(R(1-z/h)) z rho^2 drho dvarphidz#
We integrate w.r.t. #rho#:
#= mu_0 int_0^hint_0^(2pi) z [rho^3/3]_{0}^{R(1-z/h)} dvarphidz#
We assess the limits for #rho#:
#= mu_0/3 int_0^hint_0^(2pi) z (R(1-z/h))^3 dvarphidz#
#= mu_0/3 int_0^hint_0^(2pi) z R^3(1-z/h)^3 dvarphidz#
#= (mu_0 R^3)/3 int_0^hint_0^(2pi) z(-z^3/h^3 + (3 z^2)/h^2 - (3 z)/h + 1) dvarphidz#
#= (mu_0 R^3)/3 int_0^hint_0^(2pi) (-z^4/h^3 + (3 z^3)/h^2 - (3 z^2)/h + z) dvarphidz#
Now we integrate with w.r.t. #varphi#:
#= (mu_0 R^3)/3 int_0^h 2pi(-z^4/h^3 + (3 z^3)/h^2 - (3 z^2)/h + z) dz#
Lastly we integrate w.r.t. #z#:
#= (2pi mu_0 R^3)/3 int_0^h (-z^4/h^3 + (3 z^3)/h^2 - (3 z^2)/h + z) dz#
We assess the limits for #z#:
#= (2pi mu_0 R^3)/3 [(-z^5/(5h^3) + (3 z^4)/(4h^2) - (z^3)/h + z^2/2)]_{0}^{h}#
#= (2pi mu_0 R^3)/3 (-h^5/(5h^3) + (3 h^4)/(4h^2) - (h^3)/h + h^2/2)#
#= (2pi mu_0 R^3)/3 (-h^2/(5) + (3 h^2)/(4) - h^2 + h^2/2)#
#= (2pi mu_0 R^3)/3 (-(4h^2)/(20) + (15 h^2)/(20) - (20h^2)/20 + (10h^2)/20)#
#= (2pi mu_0 R^3)/3 (h^2/20)#
#= (2pi mu_0 R^3h^2)/60#
#= (pi mu_0 R^3h^2)/30#
Hence:
#=>color(green)(Z_S = (pi mu_0 R^3h^2)/30)#
