State and prove parallel axis theorem

Explanation

Parallel axis theorem: Parallel axis theorem states that the moment of inertia of a body about any axis is equal to the sum of its moment of inertia about a parallel axis through its center of mass and the product of the mass of the body and the square of the perpendicular distance between the two axes.
If `I_(C)` is the moment of inertia of the body of mass M about an axis passing through the center of mass, then the moment of inertial about a parallel axis at a distance d from it is given by the relation,
`I=I_(C)=Md^(2)`
Let us consider a rigid body as shown in figure. Its moment of inertia about an axis AB passing through the center of mass is `I_(C)` DE is another axis parallel to AB at a perpendicular distance from AB. The moment of inertia of the body about DE is 1. We attempt to get an expression for in terms of `I_(C)`. For this, let us consider a point mass on the body at position x from its center of mass.
The moment of inertia of the point mass about the axis DE is, `m (x+d)^(2)`. The moment of inertial of the whole body about DE is the summation of the above expression.
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`I= sum m(x+d)^(2)`
This equation could further be written as,
`I= sum m(x^(2)+d^(2)+2xd)`
`I=sum (mx^(2)+md^(2)+2dmx)`
`I= sum mx^(2)+sum md^(2)+2d sum mx`
Here, `sum x^(2)` is the moment of inertia of the body about the center of mass. Hence, `I_(C)=sum mx^(2)`
The term, `sum mx=0` because, x can take positive and negative values with respect to the axis AB. The summation `(sum mx)` will be zero.
Thus `I=I_(C)+ sum md^(2)=I_(C)+(sum m)d^(2)`
Here, `sum m` is the entire mass of M of the object `(sum m=M)`