I = ∑ M_{i}R_{i}^{2}

I_{central} = \frac{1}{12}mL^{2}

I_{shifted} = I_{central} + mx^{2}

= \frac{1}{12}mL^{2} + mx^{2}

I_{2−rod} = I_{central} + I_{shifted}

= (\frac{1}{12}mL^{2}) + (\frac{1}{12}mL^{2} + Mx^{2})

surface density σ = \frac{M}{A} = \frac{M}{L^{2}}

dM = σ ∗ ( area ) = σ ∗ (L ∗ dx)

\begin{aligned}dI & = (\frac{1}{12} dM L^{2}) + (dM x^{2}) \\ dI & = (\frac{1}{12}(σLdx)L^{2}) + ((σLdx)x^{2}) \\ dI & = (\frac{1}{12}σL^{3}dx) + (σLx^{2}dx) \end{aligned}

I = ∫ dm R^{2}

\begin{aligned}W & = ∫ F⃗ · dx⃗ \\ W & = \frac{1}{2}mv_{f}^{2} − \frac{1}{2}mv_{i}^{2} \\ P & = F⃗ · v⃗ \end{aligned}  \begin{aligned}W & = ∫ τ⃗ · dθ⃗ \\ W & = \frac{1}{2}Iω_{f}^{2} − \frac{1}{2}Iω_{i}^{2} \\ P & = τ⃗ · ω⃗ \end{aligned}