Consider the region D={(x,y,z):x2+y2+z2≤r2}
The region D is inside the sphere of radius r , so the region in spherical coordinates is given by,
D={(ρ,ϕ,θ)∣0≤ρ≤r,0≤ϕ≤π,0≤θ≤2π}
Now, evaluate the triple integral in spherical coordinates as,
∭DcdV=∫02π∫0π∫0rcρ2sinϕdρdϕdθ
=∫02π∫0πc[3ρ3]0rsinϕdϕdθ
=3cr3∫02π∫0πsinϕdϕdθ
=3cr3∫02π[−cosϕ]0πdθ
=3cr3∫02π(2)dθ
=32cr3∫02πdθ
=32cr3[θ]02π
=32cr3(2π)
=34πcr3
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