∭Sx2+y2+z232dV=

This is a triple integral, where we need to integrate over a 3D region S. The function being integrated is (x^2 + y^2 + z^2)^(3/2).

Since we want to integrate over a spherical region, we can use spherical coordinates. The limits of integration will be:

ρ: 0 to R (the radius of the sphere) θ: 0 to 2π (the azimuthal angle) φ: 0 to π (the polar angle)

The integrand in spherical coordinates is:

ρ^5(sinφ)^3

The Jacobian for the transformation from Cartesian to spherical coordinates is:

ρ^2sinφ

Therefore, the integral becomes:

∭S(x^2 + y^2 + z^2)^(3/2) dV = ∫0^R ∫0^π ∫0^2π ρ^5(sinφ)^3 ρ^2sinφ dθ dφ dρ

Simplifying this expression and performing the integrals, we get:

(4/15)πR^