diff --git a/Lectures_my/NumMet/2016/Lecture7/lecture7.tex b/Lectures_my/NumMet/2016/Lecture7/lecture7.tex
index 1e5beec..e2b4626 100644
--- a/Lectures_my/NumMet/2016/Lecture7/lecture7.tex
+++ b/Lectures_my/NumMet/2016/Lecture7/lecture7.tex
@@ -243,10 +243,12 @@
     \end{itemize}
     \vfill
     The integral can then be approximated as
-    \begin{equation}
-        I \approx \sum_{k=0}^{N \cdot M + 1} \omega_k f(x_k)
-    \end{equation}
-    with $x_k = a + (b - a) \frac{k}{N\cdot M}$\,.
+    \begin{align*}
+        I = \sum_{m=1}^{M} \int_{x_{m \cdot N}}^{x_{(m+1)\cdot N}} \mathrm{d}x\, f(x)
+          = \sum_{m=1}^{M} \sum_{n=0}^{N} \omega_k f(x_{m\cdot N + n})
+          = \sum_{k=0}^{K = N \cdot M} \Omega_k f(x_k)
+    \end{align*}
+    with $x_k = a + (b - a) \frac{k}{K\cdot M}$\,.
     }
     \only<2->{
     \resizebox{.9\textwidth}{!}{
@@ -299,15 +301,29 @@
     }
     }
 \end{frame}
-\begin{frame}{}
-    \begin{equation}
-        \omega_k = \begin{cases}
-            \frac{1}{6}   & k = 0 \lor k = N \cdot M + 1\\
-            \frac{4}{6}   & k\,\text{even}\\
-            \frac{2}{6}   & k\,\text{odd}
-        \end{cases}
-    \end{equation}
 
+\begin{frame}{(Closed) Newton-Cotes Weights and error estimates}
+    %
+
+    {
+    \centering
+    \renewcommand{\arraystretch}{1.2}
+    \begin{tabular}{c|c|c}
+        $\bm{N}$ & $\bm{\omega^{(N)}_k}$                             & \textbf{approx. error}\\
+        \hline
+        $1$      & $\frac{1}{2}\, \frac{1}{2}$                       & $\sim (b - a) f''(\xi)$\\
+        $2$      & $\frac{1}{6}\, \frac{4}{6}\, \frac{1}{6}$         & $\sim (b - a) f^{(4)}(\xi)$\\
+        $3$      & $\frac18\, \frac38\, \frac38\, \frac18$           & $\sim (b - a) f^{(4)}(\xi)$\\
+        $\vdots$ & $\vdots$                                          & $\vdots$\\
+    \end{tabular}
+    \renewcommand{\arraystretch}{1}
+    }
+
+    Comments and observations:
+    \begin{itemize}
+        \item $\xi \in [a, b]$
+        \item While numerical factors change, the approximation error scales only with an even derivative of the integrand!
+    \end{itemize}
 \end{frame}
 
 \newcommand{\eps}{\varepsilon}
@@ -350,7 +366,8 @@
                 I      & = 5.57396
             \end{align*}
         \end{block}
-        \begin{block}{Entire range, 7 data points}
+        \vspace{-\smallskipamount}
+        \begin{block}<2->{Entire range, 7 data points}
             ~\\
             \vspace{-2\bigskipamount}
             \begin{align*}
@@ -358,7 +375,8 @@
                 \delta_7 & = 12.26\%
             \end{align*}
         \end{block}
-        \begin{block}{Two integration, 5 data points}
+        \vspace{-\smallskipamount}
+        \begin{block}<3->{Two integration, 5 data points}
             ~\\
             \vspace{-2\bigskipamount}
             \begin{align*}
@@ -368,6 +386,46 @@
         \end{block}
     \end{column}
     \end{columns}
+    \only<4->{
+    It is benefitial to arrange the integration intervals around known special points of the integrand!
+    }
+\end{frame}
+
+\begin{frame}{Rationale for Gauss quadrature}
+    Given that we ended up (again) using polynomial interpolation to numerically solve a problem,
+    we can ask ourselves the following question:\\
+    \vfill
+    \alert{Is there a method to numerically obtain the exact integral of a polynomial of degree $2n - 1$?}
+    \vfill
+    Carl Friedrich Gau\ss{} answered in the affirmative!
+\end{frame}
+
+\begin{frame}{General idea}
+    We restrict ourselves to the interval $[a, b]$, and to integrals of the type
+    \begin{equation*}
+        I = \int_{a}^{b} \mathrm{d}x\, \omega(x) f(x)
+    \end{equation*}
+    Nomenclature:
+    \begin{itemize}
+        \item $\omega(x)$ is the weight function: $\omega(x) > 0$, integrable and has finitely many roots.
+        \item $p_N(x)$ is a polynomial of degree $N$ over $x$.
+        \item $\xi_{n}$, with $n=1,\dots,N$ are the roots of $p_N(x)$.
+    \end{itemize}
+    Orthogonality relations apply:
+    \begin{equation*}
+        \langle p_i(x), p_j(x)\rangle \equiv \int_{a}^{b} \mathrm{d}x\, \omega(x) p_i(x) p_j(x) = \delta_{ij}
+    \end{equation*}
+\end{frame}
+
+\begin{frame}{General idea cont'd}
+    We can then approximate
+    \begin{equation*}
+        I \approx \sum_{n=1}^N \alpha_n f(\xi_n)
+    \end{equation*}
+    where the coefficients $\alpha_n$ can be determined as
+    \begin{equation*}
+        \alpha_n = \int_{a}^{b} \mathrm{d}x\, \omega(x) \prod_{m=1,n\neq m}^{N} \frac{x - \xi_n}{\xi_m - \xi_n}
+    \end{equation*}
 \end{frame}
 
 \backupbegin