Answer
a) The proof is below.
b) $I =M(h^2+r_{cm}^2)$
Work Step by Step
a) We know that the Law of Cosines is:
$C^2 = A^2 + B^2 -2AB cos \gamma$
Where $\gamma$ is the angle opposite to side C.
Thus, plugging in $r_{cm}$ for A, $h$ for B, and $r$ for C, we find:
$r^2 =r_{cm}^2 + h^2 -2hr_{cm} cos \gamma$
Looking at the value of gamma, this simplifies to:
$r^2 =r_{cm}^2 + h^2 -2hr_{cm} $
b) As the book recommends in the problem, we take the integral of $r^2$ and use substitution to solve:
$I=\int r^2 dm$
$I=\int (r_{cm}^2 + h^2 -2hr_{cm}) dm$
Taking the integral, this becomes:
$I =M(h^2+r_{cm}^2)$
