91Ó°ÊÓ

$∈$is the portion of unit sphere centered at the origin of first octant.

To determine the first moment of solid w.r.t $yz$plane, formula to be used:

$M_{YZ}=∫∫{∫}_{∈}x\mathrm{ÒÏ}(x,y,z)dxdydz$

Here, $k$is uniform density of $∈$

$⇒\mathrm{ÒÏ}\left(x,y,z\right)=k$

We know the relation:

$x=\mathrm{ÒÏ}\sin \mathrm{Ï•}\cos \mathrm{θ},y=\mathrm{ÒÏ}\sin \mathrm{Ï•}\sin \mathrm{θ},z=\mathrm{ÒÏ}\cos \mathrm{Ï•}$

Here, $\mathrm{ÒÏ}=\sqrt{x^2+y^2+z^2},\tan \mathrm{θ}=\frac{y}{z},\cos \mathrm{Ï•}=\frac{z}{\mathrm{ÒÏ}}$

Also $dxdydz={\mathrm{ÒÏ}}^2\sin \mathrm{Ï•}d\mathrm{ÒÏ}d\mathrm{Ï•}d\mathrm{θ}$

$0<\mathrm{θ}<\frac{\mathrm{Ï€}}{2},0<\mathrm{Ï•}<\frac{\mathrm{Ï€}}{2},0<\mathrm{ÒÏ}<1$are limits of spherical coordinates $\mathrm{ÒÏ},\mathrm{Ï•},\mathrm{θ}$

First moment of $∈=∫∫{∫}_{V}x\mathrm{ÒÏ}\left(x,y,z\right)dxdydz$

Converting into spherical coordinates

$={∫}_{\mathrm{Ï•}=0}^{\frac{\mathrm{Ï€}}{2}}{∫}_{\mathrm{θ}=0}^{\frac{\mathrm{Ï€}}{2}}{∫}_{\mathrm{ÒÏ}=0}^1{\mathrm{ÒÏ}}^2\sin \mathrm{Ï•}\left(\mathrm{ÒÏ}\sin \mathrm{Ï•}\cos \mathrm{θ}\right)kd\mathrm{ÒÏ}d\mathrm{Ï•}d\mathrm{θ}$

$=k\left[{∫}_{\mathrm{Ï•}=0}^{\frac{\mathrm{Ï€}}{2}}{\sin }^2\mathrm{Ï•}d\mathrm{Ï•}\right]×\left[{∫}_{\mathrm{θ}=0}^{\frac{\mathrm{Ï€}}{2}}\cos \mathrm{θ}d\mathrm{θ}\right]×\left[{∫}_{\mathrm{ÒÏ}=0}^1{\mathrm{ÒÏ}}^{3}d\mathrm{ÒÏ}\right]$

Solving the integral

$=k\left(\frac{1}{2}×\frac{\mathrm{Ï€}}{2}\right){\left(\frac{{\mathrm{ÒÏ}}^4}{4}\right)}_0^1{\left(\sin \mathrm{θ}\right)}_0^{\frac{\mathrm{Ï€}}{2}}$

By application of limits, we get

$M_{YZ}=k\left(\frac{\mathrm{Ï€}}{4}\right)\left(\frac{1}{4}\right)\left(1\right)$

Hence, first moment is$M_{YZ}=\frac{\mathrm{Ï€°ì}}{16}$