diff --git a/bachelorproject.pdf b/bachelorproject.pdf
index d185662..417cff0 100644
Binary files a/bachelorproject.pdf and b/bachelorproject.pdf differ
diff --git a/sections/appendix.tex b/sections/appendix.tex
index 454623f..ef4ab9f 100644
--- a/sections/appendix.tex
+++ b/sections/appendix.tex
@@ -46,11 +46,19 @@ it will be useful to simplify the result of the integral.
 \end{equation}
 
 \newpage
-\paragraph{Moment of inertia of sub-triangle of arbitrary polygon}
+\paragraph{Moment of inertia of sub-triangle of arbitrary polygon} Recall
+equation \ref{eq:r} defines
+$$ \vec r = \alpha \vv{CA} + \beta \alpha \vv{AB} $$
 \begin{equation}
 	\label{eq:subtriangle_arbitrary_moment_long}
 	\begin{split}
-		I_{T_i} = \rho \int_0^1 \int_0^1 \vec r^2 hb \alpha  \diff \alpha \diff \beta\\
-		&=
+		I_{T_i} &= \rho \int_0^1 \int_0^1 \vec r^2 hb \alpha  \diff \alpha \diff \beta\\
+		&=\rho hb\int_0^1 \int_0^1 \left(\alpha \vv{CA} + \beta \alpha \vv{AB}\right)^2 \alpha  \diff \alpha \diff \beta\\
+		&=\rho hb\int_0^1 \int_0^1 \left(\alpha^2 \vv{CA}^2 + 2 \alpha^2 \beta \vv{AB} \cdot \vv{CA} + \alpha^2 \beta^2 \vv{AB}^2\right) \alpha  \diff \alpha \diff \beta\\
+		&=\rho hb\int_0^1 \int_0^1 \alpha^3 \left(\vv{CA}^2 + 2 \beta \vv{AB} \cdot \vv{CA} + \beta^2 \vv{AB}^2\right) \diff \alpha \diff \beta\\
+		&=\rho hb\int_0^1 \Biggl[\frac{1}{4} \alpha^4 \left(\vv{CA}^2 + 2 \beta \vv{AB} \cdot \vv{CA} + \beta^2 \vv{AB}^2\right) \Biggr]_0^1\diff \beta\\
+		&= \frac{\rho hb}{4}\int_0^1 \beta^2 \vv{AB}^2 + 2 \beta \vv{AB} \cdot \vv{CA} + \vv{CA}^2 \diff \beta\\
+		&= \frac{\rho hb}{4} \Biggl[\frac{1}{3} \beta^3 \vv{AB}^2 + \beta^2 \vv{AB} \cdot \vv{CA} + \beta\vv{CA}^2 \Biggr]_0^1\\
+		&= \frac{\rho hb}{4} \left(\frac{1}{3}\vv{AB}^2 + \vv{AB} \cdot \vv{CA} + \vv{CA}^2\right) \\
 	\end{split}
 \end{equation}