Correct option is (b).
Reason:
Given ,c,r are constants.
Assume radius is R instead of r for the time being,then replace R by r in the final result.
D is the region described as D={(x,y,z):x2+y2+z2≤r2} ,clearly D contains all the points lies inside and surface of the sphere whose radius is r .
Thus,
I=∫∫∫DcdV=c∫∫∫DdV Let's draw the elemental volume
Hence,
I=c∫∫∫DdV=c∫0R∫0π∫02πr2sin(θ)dθdϕdr⟹I=c∫0R(r2∫0π(sin(θ)∫02πdϕ)dθ)dr=c34πR3 Therefore the final answer is
I=c34πr3
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