\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus
Please login to create an exam or a quiz.
Use 3 rectangles to approximate \(\displaystyle \int\limits_{1}^{8} \ln{\left(x \right)}\, dx\) using the midpoint. Draw a graph showing the sum.
\(\displaystyle \Delta x = \frac{b-a}{n}=\frac{7}{3}\) and \(\displaystyle x_i = a+i\Delta x +\frac{\Delta x}{2} = \frac{7 i}{3} + \frac{13}{6}\). Drawing a graph gives:
\begin{tikzpicture}[yscale=2]
\draw[very thick, color=blue, opacity=0.3, xstep=1, ystep=1] (-1,-1) grid (9,3.10000000000000);
\draw[latex-latex, very thick] (-1.4,0)--(9.4,0) node[right]{$x$};
\draw[latex-latex, very thick] (0,-1.1)--(0,3.2) node[above]{$y$};
\foreach \x in { -1,1,...,9 }{
\pgfmathsetmacro{\xtick}{\x != 0 ? "\x" : ""}
\draw (\x, .2) -- (\x,-.2) node[below]{$\xtick$};
}
\foreach \y in { -1,-0.5,...,3.10000000000000 }{
\pgfmathsetmacro{\ytick}{\y != 0 ? "\y" : ""}
\draw (.2, \y) -- (-.2, \y) node[left]{$\ytick$};
}
\draw[thick, color=blue, domain={1.0}:{8.0}, samples=200, samples =350] plot(\x, {ln((\x))});
\draw[fill=blue, opacity=0.3](1.0000, 0) rectangle (3.3333, 0.77319);
\draw[thick](1.0000, 0) rectangle (3.3333, 0.77319);
\draw[fill=blue, opacity=0.3](3.3333, 0) rectangle (5.6667, 1.5041);
\draw[thick](3.3333, 0) rectangle (5.6667, 1.5041);
\draw[fill=blue, opacity=0.3](5.6667, 0) rectangle (8.0000, 1.9218);
\draw[thick](5.6667, 0) rectangle (8.0000, 1.9218);
\end{tikzpicture}
\begin{question}Use 3 rectangles to approximate $\int\limits_{1}^{8} \ln{\left(x \right)}\, dx$ using the midpoint. Draw a graph showing the sum.
\soln{9cm}{$\Delta x = \frac{b-a}{n}=\frac{7}{3}$ and $x_i = a+i\Delta x +\frac{\Delta x}{2} = \frac{7 i}{3} + \frac{13}{6}$. Drawing a graph gives:
\begin{center}\begin{tikzpicture}[yscale=2]
\draw[very thick, color=blue, opacity=0.3, xstep=1, ystep=1] (-1,-1) grid (9,3.10000000000000);
\draw[latex-latex, very thick] (-1.4,0)--(9.4,0) node[right]{$x$};
\draw[latex-latex, very thick] (0,-1.1)--(0,3.2) node[above]{$y$};
\foreach \x in { -1,1,...,9 }{
\pgfmathsetmacro{\xtick}{\x != 0 ? "\x" : ""}
\draw (\x, .2) -- (\x,-.2) node[below]{$\xtick$};
}
\foreach \y in { -1,-0.5,...,3.10000000000000 }{
\pgfmathsetmacro{\ytick}{\y != 0 ? "\y" : ""}
\draw (.2, \y) -- (-.2, \y) node[left]{$\ytick$};
}
\draw[thick, color=blue, domain={1.0}:{8.0}, samples=200, samples =350] plot(\x, {ln((\x))});
\draw[fill=blue, opacity=0.3](1.0000, 0) rectangle (3.3333, 0.77319);
\draw[thick](1.0000, 0) rectangle (3.3333, 0.77319);
\draw[fill=blue, opacity=0.3](3.3333, 0) rectangle (5.6667, 1.5041);
\draw[thick](3.3333, 0) rectangle (5.6667, 1.5041);
\draw[fill=blue, opacity=0.3](5.6667, 0) rectangle (8.0000, 1.9218);
\draw[thick](5.6667, 0) rectangle (8.0000, 1.9218);
\end{tikzpicture}
\end{center}
The sum is $\sum_{i=0}^{2} \frac{7 \ln{\left(\frac{7 i}{3} + \frac{13}{6} \right)}}{3} = 9.7979$. }
\end{question}
\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}
\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}
\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}
\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
\fi%
}%
\everymath{\displaystyle}
\ShowSolutiontrue
\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Use 3 rectangles to approximate <img class="equation_image" title=" \displaystyle \int\limits_{1}^{8} \ln{\left(x \right)}\, dx " src="/equation_images/%20%5Cdisplaystyle%20%5Cint%5Climits_%7B1%7D%5E%7B8%7D%20%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%5C%2C%20dx%20" alt="LaTeX: \displaystyle \int\limits_{1}^{8} \ln{\left(x \right)}\, dx " data-equation-content=" \displaystyle \int\limits_{1}^{8} \ln{\left(x \right)}\, dx " /> using the midpoint. Draw a graph showing the sum. </p> </p><p> <p> <img class="equation_image" title=" \displaystyle \Delta x = \frac{b-a}{n}=\frac{7}{3} " src="/equation_images/%20%5Cdisplaystyle%20%5CDelta%20x%20%3D%20%5Cfrac%7Bb-a%7D%7Bn%7D%3D%5Cfrac%7B7%7D%7B3%7D%20" alt="LaTeX: \displaystyle \Delta x = \frac{b-a}{n}=\frac{7}{3} " data-equation-content=" \displaystyle \Delta x = \frac{b-a}{n}=\frac{7}{3} " /> and <img class="equation_image" title=" \displaystyle x_i = a+i\Delta x +\frac{\Delta x}{2} = \frac{7 i}{3} + \frac{13}{6} " src="/equation_images/%20%5Cdisplaystyle%20x_i%20%3D%20a%2Bi%5CDelta%20x%20%2B%5Cfrac%7B%5CDelta%20x%7D%7B2%7D%20%3D%20%5Cfrac%7B7%20i%7D%7B3%7D%20%2B%20%5Cfrac%7B13%7D%7B6%7D%20" alt="LaTeX: \displaystyle x_i = a+i\Delta x +\frac{\Delta x}{2} = \frac{7 i}{3} + \frac{13}{6} " data-equation-content=" \displaystyle x_i = a+i\Delta x +\frac{\Delta x}{2} = \frac{7 i}{3} + \frac{13}{6} " /> . Drawing a graph gives:
<center><?xml version="1.0" encoding="UTF-8"?>
<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="319.672pt" height="258.521pt" viewBox="0 0 319.672 258.521" version="1.1">
<defs>
<g>
<symbol overflow="visible" id="glyph0-0">
<path style="stroke:none;" d=""/>
</symbol>
<symbol overflow="visible" id="glyph0-1">
<path style="stroke:none;" d="M 3.328125 -3.015625 C 3.390625 -3.265625 3.625 -4.1875 4.3125 -4.1875 C 4.359375 -4.1875 4.609375 -4.1875 4.8125 -4.0625 C 4.53125 -4 4.34375 -3.765625 4.34375 -3.515625 C 4.34375 -3.359375 4.453125 -3.171875 4.71875 -3.171875 C 4.9375 -3.171875 5.25 -3.34375 5.25 -3.75 C 5.25 -4.265625 4.671875 -4.40625 4.328125 -4.40625 C 3.75 -4.40625 3.40625 -3.875 3.28125 -3.65625 C 3.03125 -4.3125 2.5 -4.40625 2.203125 -4.40625 C 1.171875 -4.40625 0.59375 -3.125 0.59375 -2.875 C 0.59375 -2.765625 0.703125 -2.765625 0.71875 -2.765625 C 0.796875 -2.765625 0.828125 -2.796875 0.84375 -2.875 C 1.1875 -3.9375 1.84375 -4.1875 2.1875 -4.1875 C 2.375 -4.1875 2.71875 -4.09375 2.71875 -3.515625 C 2.71875 -3.203125 2.546875 -2.546875 2.1875 -1.140625 C 2.03125 -0.53125 1.671875 -0.109375 1.234375 -0.109375 C 1.171875 -0.109375 0.953125 -0.109375 0.734375 -0.234375 C 0.984375 -0.296875 1.203125 -0.5 1.203125 -0.78125 C 1.203125 -1.046875 0.984375 -1.125 0.84375 -1.125 C 0.53125 -1.125 0.296875 -0.875 0.296875 -0.546875 C 0.296875 -0.09375 0.78125 0.109375 1.21875 0.109375 C 1.890625 0.109375 2.25 -0.59375 2.265625 -0.640625 C 2.390625 -0.28125 2.75 0.109375 3.34375 0.109375 C 4.375 0.109375 4.9375 -1.171875 4.9375 -1.421875 C 4.9375 -1.53125 4.859375 -1.53125 4.828125 -1.53125 C 4.734375 -1.53125 4.71875 -1.484375 4.6875 -1.421875 C 4.359375 -0.34375 3.6875 -0.109375 3.375 -0.109375 C 2.984375 -0.109375 2.828125 -0.421875 2.828125 -0.765625 C 2.828125 -0.984375 2.875 -1.203125 2.984375 -1.640625 Z M 3.328125 -3.015625 "/>
</symbol>
<symbol overflow="visible" id="glyph0-2">
<path style="stroke:none;" d="M 4.84375 -3.796875 C 4.890625 -3.9375 4.890625 -3.953125 4.890625 -4.03125 C 4.890625 -4.203125 4.75 -4.296875 4.59375 -4.296875 C 4.5 -4.296875 4.34375 -4.234375 4.25 -4.09375 C 4.234375 -4.03125 4.140625 -3.734375 4.109375 -3.546875 C 4.03125 -3.296875 3.96875 -3.015625 3.90625 -2.75 L 3.453125 -0.953125 C 3.421875 -0.8125 2.984375 -0.109375 2.328125 -0.109375 C 1.828125 -0.109375 1.71875 -0.546875 1.71875 -0.921875 C 1.71875 -1.375 1.890625 -2 2.21875 -2.875 C 2.375 -3.28125 2.421875 -3.390625 2.421875 -3.59375 C 2.421875 -4.03125 2.109375 -4.40625 1.609375 -4.40625 C 0.65625 -4.40625 0.296875 -2.953125 0.296875 -2.875 C 0.296875 -2.765625 0.390625 -2.765625 0.40625 -2.765625 C 0.515625 -2.765625 0.515625 -2.796875 0.5625 -2.953125 C 0.84375 -3.890625 1.234375 -4.1875 1.578125 -4.1875 C 1.65625 -4.1875 1.828125 -4.1875 1.828125 -3.875 C 1.828125 -3.625 1.71875 -3.359375 1.65625 -3.171875 C 1.25 -2.109375 1.078125 -1.546875 1.078125 -1.078125 C 1.078125 -0.1875 1.703125 0.109375 2.296875 0.109375 C 2.6875 0.109375 3.015625 -0.0625 3.296875 -0.34375 C 3.171875 0.171875 3.046875 0.671875 2.65625 1.203125 C 2.390625 1.53125 2.015625 1.828125 1.5625 1.828125 C 1.421875 1.828125 0.96875 1.796875 0.796875 1.40625 C 0.953125 1.40625 1.09375 1.40625 1.21875 1.28125 C 1.328125 1.203125 1.421875 1.0625 1.421875 0.875 C 1.421875 0.5625 1.15625 0.53125 1.0625 0.53125 C 0.828125 0.53125 0.5 0.6875 0.5 1.171875 C 0.5 1.671875 0.9375 2.046875 1.5625 2.046875 C 2.578125 2.046875 3.609375 1.140625 3.890625 0.015625 Z M 4.84375 -3.796875 "/>
</symbol>
<symbol overflow="visible" id="glyph0-3">
<path style="stroke:none;" d="M 1.90625 -0.53125 C 1.90625 -0.8125 1.671875 -1.0625 1.390625 -1.0625 C 1.09375 -1.0625 0.859375 -0.8125 0.859375 -0.53125 C 0.859375 -0.234375 1.09375 0 1.390625 0 C 1.671875 0 1.90625 -0.234375 1.90625 -0.53125 Z M 1.90625 -0.53125 "/>
</symbol>
<symbol overflow="visible" id="glyph1-0">
<path style="stroke:none;" d=""/>
</symbol>
<symbol overflow="visible" id="glyph1-1">
<path style="stroke:none;" d="M 6.5625 -2.296875 C 6.734375 -2.296875 6.921875 -2.296875 6.921875 -2.5 C 6.921875 -2.6875 6.734375 -2.6875 6.5625 -2.6875 L 1.171875 -2.6875 C 1 -2.6875 0.828125 -2.6875 0.828125 -2.5 C 0.828125 -2.296875 1 -2.296875 1.171875 -2.296875 Z M 6.5625 -2.296875 "/>
</symbol>
<symbol overflow="visible" id="glyph2-0">
<path style="stroke:none;" d=""/>
</symbol>
<symbol overflow="visible" id="glyph2-1">
<path style="stroke:none;" d="M 2.9375 -6.375 C 2.9375 -6.625 2.9375 -6.640625 2.703125 -6.640625 C 2.078125 -6 1.203125 -6 0.890625 -6 L 0.890625 -5.6875 C 1.09375 -5.6875 1.671875 -5.6875 2.1875 -5.953125 L 2.1875 -0.78125 C 2.1875 -0.421875 2.15625 -0.3125 1.265625 -0.3125 L 0.953125 -0.3125 L 0.953125 0 C 1.296875 -0.03125 2.15625 -0.03125 2.5625 -0.03125 C 2.953125 -0.03125 3.828125 -0.03125 4.171875 0 L 4.171875 -0.3125 L 3.859375 -0.3125 C 2.953125 -0.3125 2.9375 -0.421875 2.9375 -0.78125 Z M 2.9375 -6.375 "/>
</symbol>
<symbol overflow="visible" id="glyph2-2">
<path style="stroke:none;" d="M 2.890625 -3.515625 C 3.703125 -3.78125 4.28125 -4.46875 4.28125 -5.265625 C 4.28125 -6.078125 3.40625 -6.640625 2.453125 -6.640625 C 1.453125 -6.640625 0.6875 -6.046875 0.6875 -5.28125 C 0.6875 -4.953125 0.90625 -4.765625 1.203125 -4.765625 C 1.5 -4.765625 1.703125 -4.984375 1.703125 -5.28125 C 1.703125 -5.765625 1.234375 -5.765625 1.09375 -5.765625 C 1.390625 -6.265625 2.046875 -6.390625 2.40625 -6.390625 C 2.828125 -6.390625 3.375 -6.171875 3.375 -5.28125 C 3.375 -5.15625 3.34375 -4.578125 3.09375 -4.140625 C 2.796875 -3.65625 2.453125 -3.625 2.203125 -3.625 C 2.125 -3.609375 1.890625 -3.59375 1.8125 -3.59375 C 1.734375 -3.578125 1.671875 -3.5625 1.671875 -3.46875 C 1.671875 -3.359375 1.734375 -3.359375 1.90625 -3.359375 L 2.34375 -3.359375 C 3.15625 -3.359375 3.53125 -2.6875 3.53125 -1.703125 C 3.53125 -0.34375 2.84375 -0.0625 2.40625 -0.0625 C 1.96875 -0.0625 1.21875 -0.234375 0.875 -0.8125 C 1.21875 -0.765625 1.53125 -0.984375 1.53125 -1.359375 C 1.53125 -1.71875 1.265625 -1.921875 0.984375 -1.921875 C 0.734375 -1.921875 0.421875 -1.78125 0.421875 -1.34375 C 0.421875 -0.4375 1.34375 0.21875 2.4375 0.21875 C 3.65625 0.21875 4.5625 -0.6875 4.5625 -1.703125 C 4.5625 -2.515625 3.921875 -3.296875 2.890625 -3.515625 Z M 2.890625 -3.515625 "/>
</symbol>
<symbol overflow="visible" id="glyph2-3">
<path style="stroke:none;" d="M 4.46875 -2 C 4.46875 -3.1875 3.65625 -4.1875 2.578125 -4.1875 C 2.109375 -4.1875 1.671875 -4.03125 1.3125 -3.671875 L 1.3125 -5.625 C 1.515625 -5.5625 1.84375 -5.5 2.15625 -5.5 C 3.390625 -5.5 4.09375 -6.40625 4.09375 -6.53125 C 4.09375 -6.59375 4.0625 -6.640625 3.984375 -6.640625 C 3.984375 -6.640625 3.953125 -6.640625 3.90625 -6.609375 C 3.703125 -6.515625 3.21875 -6.3125 2.546875 -6.3125 C 2.15625 -6.3125 1.6875 -6.390625 1.21875 -6.59375 C 1.140625 -6.625 1.125 -6.625 1.109375 -6.625 C 1 -6.625 1 -6.546875 1 -6.390625 L 1 -3.4375 C 1 -3.265625 1 -3.1875 1.140625 -3.1875 C 1.21875 -3.1875 1.234375 -3.203125 1.28125 -3.265625 C 1.390625 -3.421875 1.75 -3.96875 2.5625 -3.96875 C 3.078125 -3.96875 3.328125 -3.515625 3.40625 -3.328125 C 3.5625 -2.953125 3.59375 -2.578125 3.59375 -2.078125 C 3.59375 -1.71875 3.59375 -1.125 3.34375 -0.703125 C 3.109375 -0.3125 2.734375 -0.0625 2.28125 -0.0625 C 1.5625 -0.0625 0.984375 -0.59375 0.8125 -1.171875 C 0.84375 -1.171875 0.875 -1.15625 0.984375 -1.15625 C 1.3125 -1.15625 1.484375 -1.40625 1.484375 -1.640625 C 1.484375 -1.890625 1.3125 -2.140625 0.984375 -2.140625 C 0.84375 -2.140625 0.5 -2.0625 0.5 -1.609375 C 0.5 -0.75 1.1875 0.21875 2.296875 0.21875 C 3.453125 0.21875 4.46875 -0.734375 4.46875 -2 Z M 4.46875 -2 "/>
</symbol>
<symbol overflow="visible" id="glyph2-4">
<path style="stroke:none;" d="M 4.75 -6.078125 C 4.828125 -6.1875 4.828125 -6.203125 4.828125 -6.421875 L 2.40625 -6.421875 C 1.203125 -6.421875 1.171875 -6.546875 1.140625 -6.734375 L 0.890625 -6.734375 L 0.5625 -4.6875 L 0.8125 -4.6875 C 0.84375 -4.84375 0.921875 -5.46875 1.0625 -5.59375 C 1.125 -5.65625 1.90625 -5.65625 2.03125 -5.65625 L 4.09375 -5.65625 C 3.984375 -5.5 3.203125 -4.40625 2.984375 -4.078125 C 2.078125 -2.734375 1.75 -1.34375 1.75 -0.328125 C 1.75 -0.234375 1.75 0.21875 2.21875 0.21875 C 2.671875 0.21875 2.671875 -0.234375 2.671875 -0.328125 L 2.671875 -0.84375 C 2.671875 -1.390625 2.703125 -1.9375 2.78125 -2.46875 C 2.828125 -2.703125 2.953125 -3.5625 3.40625 -4.171875 Z M 4.75 -6.078125 "/>
</symbol>
<symbol overflow="visible" id="glyph2-5">
<path style="stroke:none;" d="M 3.65625 -3.171875 L 3.65625 -2.84375 C 3.65625 -0.515625 2.625 -0.0625 2.046875 -0.0625 C 1.875 -0.0625 1.328125 -0.078125 1.0625 -0.421875 C 1.5 -0.421875 1.578125 -0.703125 1.578125 -0.875 C 1.578125 -1.1875 1.34375 -1.328125 1.125 -1.328125 C 0.96875 -1.328125 0.671875 -1.25 0.671875 -0.859375 C 0.671875 -0.1875 1.203125 0.21875 2.046875 0.21875 C 3.34375 0.21875 4.5625 -1.140625 4.5625 -3.28125 C 4.5625 -5.96875 3.40625 -6.640625 2.515625 -6.640625 C 1.96875 -6.640625 1.484375 -6.453125 1.0625 -6.015625 C 0.640625 -5.5625 0.421875 -5.140625 0.421875 -4.390625 C 0.421875 -3.15625 1.296875 -2.171875 2.40625 -2.171875 C 3.015625 -2.171875 3.421875 -2.59375 3.65625 -3.171875 Z M 2.421875 -2.40625 C 2.265625 -2.40625 1.796875 -2.40625 1.5 -3.03125 C 1.3125 -3.40625 1.3125 -3.890625 1.3125 -4.390625 C 1.3125 -4.921875 1.3125 -5.390625 1.53125 -5.765625 C 1.796875 -6.265625 2.171875 -6.390625 2.515625 -6.390625 C 2.984375 -6.390625 3.3125 -6.046875 3.484375 -5.609375 C 3.59375 -5.28125 3.640625 -4.65625 3.640625 -4.203125 C 3.640625 -3.375 3.296875 -2.40625 2.421875 -2.40625 Z M 2.421875 -2.40625 "/>
</symbol>
<symbol overflow="visible" id="glyph2-6">
<path style="stroke:none;" d="M 4.578125 -3.1875 C 4.578125 -3.984375 4.53125 -4.78125 4.1875 -5.515625 C 3.734375 -6.484375 2.90625 -6.640625 2.5 -6.640625 C 1.890625 -6.640625 1.171875 -6.375 0.75 -5.453125 C 0.4375 -4.765625 0.390625 -3.984375 0.390625 -3.1875 C 0.390625 -2.4375 0.421875 -1.546875 0.84375 -0.78125 C 1.265625 0.015625 2 0.21875 2.484375 0.21875 C 3.015625 0.21875 3.78125 0.015625 4.21875 -0.9375 C 4.53125 -1.625 4.578125 -2.40625 4.578125 -3.1875 Z M 2.484375 0 C 2.09375 0 1.5 -0.25 1.328125 -1.203125 C 1.21875 -1.796875 1.21875 -2.71875 1.21875 -3.3125 C 1.21875 -3.953125 1.21875 -4.609375 1.296875 -5.140625 C 1.484375 -6.328125 2.234375 -6.421875 2.484375 -6.421875 C 2.8125 -6.421875 3.46875 -6.234375 3.65625 -5.25 C 3.765625 -4.6875 3.765625 -3.9375 3.765625 -3.3125 C 3.765625 -2.5625 3.765625 -1.890625 3.65625 -1.25 C 3.5 -0.296875 2.9375 0 2.484375 0 Z M 2.484375 0 "/>
</symbol>
<symbol overflow="visible" id="glyph2-7">
<path style="stroke:none;" d="M 1.265625 -0.765625 L 2.328125 -1.796875 C 3.875 -3.171875 4.46875 -3.703125 4.46875 -4.703125 C 4.46875 -5.84375 3.578125 -6.640625 2.359375 -6.640625 C 1.234375 -6.640625 0.5 -5.71875 0.5 -4.828125 C 0.5 -4.28125 1 -4.28125 1.03125 -4.28125 C 1.203125 -4.28125 1.546875 -4.390625 1.546875 -4.8125 C 1.546875 -5.0625 1.359375 -5.328125 1.015625 -5.328125 C 0.9375 -5.328125 0.921875 -5.328125 0.890625 -5.3125 C 1.109375 -5.96875 1.65625 -6.328125 2.234375 -6.328125 C 3.140625 -6.328125 3.5625 -5.515625 3.5625 -4.703125 C 3.5625 -3.90625 3.078125 -3.125 2.515625 -2.5 L 0.609375 -0.375 C 0.5 -0.265625 0.5 -0.234375 0.5 0 L 4.203125 0 L 4.46875 -1.734375 L 4.234375 -1.734375 C 4.171875 -1.4375 4.109375 -1 4 -0.84375 C 3.9375 -0.765625 3.28125 -0.765625 3.0625 -0.765625 Z M 1.265625 -0.765625 "/>
</symbol>
</g>
</defs>
<g id="surface1">
<path style="fill:none;stroke-width:1.19553;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,100%);stroke-opacity:0.3;stroke-miterlimit:10;" d="M -28.3455 -56.692219 L 255.119344 -56.692219 M -28.3455 -0.0008125 L 255.119344 -0.0008125 M -28.3455 56.6945 L 255.119344 56.6945 M -28.3455 113.385906 L 255.119344 113.385906 M -28.3455 170.081219 L 255.119344 170.081219 M -28.3455 -56.692219 L -28.3455 175.749187 M -0.00175 -56.692219 L -0.00175 175.749187 M 28.345906 -56.692219 L 28.345906 175.749187 M 56.693562 -56.692219 L 56.693562 175.749187 M 85.041219 -56.692219 L 85.041219 175.749187 M 113.388875 -56.692219 L 113.388875 175.749187 M 141.732625 -56.692219 L 141.732625 175.749187 M 170.080281 -56.692219 L 170.080281 175.749187 M 198.427937 -56.692219 L 198.427937 175.749187 M 226.775594 -56.692219 L 226.775594 175.749187 M 255.111531 -56.692219 L 255.111531 175.749187 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<path style="fill:none;stroke-width:1.19553;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -33.947063 -0.0008125 L 260.720906 -0.0008125 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 0.597656 194.882812 C 2.296875 195.199219 5.0625 196.15625 6.972656 197.273438 L 6.972656 192.492188 C 5.0625 193.605469 2.296875 194.5625 0.597656 194.882812 "/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 306.742188 194.882812 C 305.042969 194.5625 302.28125 193.605469 300.367188 192.492188 L 300.367188 197.273438 C 302.28125 196.15625 305.042969 195.199219 306.742188 194.882812 "/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-1" x="310.657" y="197.027"/>
</g>
<path style="fill:none;stroke-width:1.19553;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -0.00175 -56.625813 L -0.00175 175.682781 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 40.28125 257.246094 C 40.601562 255.546875 41.558594 252.78125 42.675781 250.871094 L 37.890625 250.871094 C 39.007812 252.78125 39.964844 255.546875 40.28125 257.246094 "/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 40.28125 13.460938 C 39.964844 15.164062 39.007812 17.925781 37.890625 19.839844 L 42.675781 19.839844 C 41.558594 17.925781 40.601562 15.164062 40.28125 13.460938 "/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-2" x="37.662" y="7.61"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -28.3455 11.339031 L -28.3455 -11.33675 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="5.571" y="216.161"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="13.32" y="216.161"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 28.345906 11.339031 L 28.345906 -11.33675 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="66.138" y="216.161"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 85.041219 11.339031 L 85.041219 -11.33675 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-2" x="122.831" y="216.161"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 141.732625 11.339031 L 141.732625 -11.33675 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-3" x="179.524" y="216.161"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 198.427937 11.339031 L 198.427937 -11.33675 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-4" x="236.217" y="216.161"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 255.119344 11.339031 L 255.119344 -11.33675 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-5" x="292.91" y="216.161"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 5.670125 -56.692219 L -5.669719 -56.692219 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="18.364" y="254.37"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="26.112" y="254.37"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 5.670125 -28.348469 L -5.669719 -28.348469 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="10.615" y="226.024"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-6" x="18.364" y="226.024"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-3" x="23.345" y="226.024"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-3" x="26.112" y="226.024"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 5.670125 -0.0008125 L -5.669719 -0.0008125 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 5.670125 28.346844 L -5.669719 28.346844 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-6" x="18.364" y="169.746"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-3" x="23.345" y="169.746"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-3" x="26.112" y="169.746"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 5.670125 56.6945 L -5.669719 56.6945 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="26.112" y="141.4"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 5.670125 85.042156 L -5.669719 85.042156 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="18.364" y="113.053"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-3" x="23.345" y="113.053"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-3" x="26.112" y="113.053"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 5.670125 113.385906 L -5.669719 113.385906 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-7" x="26.112" y="84.707"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 5.670125 141.733562 L -5.669719 141.733562 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-7" x="18.364" y="56.36"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-3" x="23.345" y="56.36"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-3" x="26.112" y="56.36"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 5.670125 170.081219 L -5.669719 170.081219 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-2" x="26.112" y="28.014"/>
</g>
<path style="fill:none;stroke-width:0.79701;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,100%);stroke-opacity:1;stroke-miterlimit:10;" d="M 28.345906 -0.0008125 L 28.916219 1.124187 L 29.482625 2.229656 L 30.052937 3.311687 L 30.619344 4.370281 L 31.189656 5.417156 L 31.756062 6.440594 L 32.326375 7.4445 L 32.892781 8.432781 L 33.463094 9.405437 L 34.0295 10.358562 L 34.599812 11.299969 L 35.166219 12.221844 L 35.736531 13.132 L 36.302937 14.026531 L 36.87325 14.905437 L 37.439656 15.776531 L 38.009969 16.628094 L 38.576375 17.467937 L 39.146687 18.296062 L 39.713094 19.116375 L 40.283406 19.921062 L 40.849812 20.714031 L 41.420125 21.495281 L 41.986531 22.268719 L 42.556844 23.030437 L 43.12325 23.784344 L 43.693562 24.526531 L 44.259969 25.257 L 44.830281 25.983562 L 45.396687 26.6945 L 45.967 27.397625 L 46.533406 28.10075 L 47.103719 28.78825 L 47.670125 29.467937 L 48.240437 30.135906 L 48.806844 30.803875 L 49.377156 31.456219 L 49.943562 32.108562 L 50.513875 32.749187 L 51.080281 33.382 L 51.650594 34.007 L 52.217 34.632 L 52.787312 35.241375 L 53.353719 35.85075 L 53.924031 36.448406 L 54.490437 37.046062 L 55.06075 37.632 L 55.627156 38.214031 L 56.197469 38.78825 L 56.763875 39.366375 L 57.334187 39.932781 L 57.900594 40.491375 L 58.470906 41.046062 L 59.037312 41.592937 L 59.607625 42.135906 L 60.174031 42.674969 L 60.744344 43.206219 L 61.31075 43.733562 L 61.881062 44.257 L 62.447469 44.776531 L 63.017781 45.28825 L 63.584187 45.799969 L 64.1545 46.303875 L 64.720906 46.803875 L 65.291219 47.299969 L 65.857625 47.792156 L 66.427937 48.276531 L 66.994344 48.760906 L 67.564656 49.241375 L 68.131062 49.714031 L 68.701375 50.186687 L 69.267781 50.655437 L 69.838094 51.116375 L 70.4045 51.577312 L 70.974812 52.034344 L 71.541219 52.483562 L 72.111531 52.932781 L 72.677937 53.378094 L 73.24825 53.8195 L 73.814656 54.257 L 74.384969 54.6945 L 74.951375 55.124187 L 75.521687 55.549969 L 76.088094 55.979656 L 76.658406 56.397625 L 77.224812 56.815594 L 77.795125 57.233562 L 78.361531 57.647625 L 78.931844 58.057781 L 79.49825 58.464031 L 80.068562 58.862469 L 80.634969 59.268719 L 81.205281 59.66325 L 81.771687 60.057781 L 82.342 60.452312 L 82.908406 60.842937 L 83.478719 61.229656 L 84.045125 61.612469 L 84.615437 61.995281 L 85.181844 62.378094 L 85.752156 62.753094 L 86.318562 63.128094 L 86.888875 63.499187 L 87.455281 63.870281 L 88.025594 64.233562 L 88.592 64.60075 L 89.162312 64.960125 L 89.728719 65.323406 L 90.299031 65.682781 L 90.865437 66.03825 L 91.43575 66.385906 L 92.002156 66.745281 L 92.572469 67.092937 L 93.138875 67.436687 L 93.709187 67.780437 L 94.275594 68.128094 L 94.845906 68.467937 L 95.412312 68.803875 L 95.982625 69.139812 L 96.549031 69.479656 L 97.119344 69.811687 L 97.68575 70.139812 L 98.256062 70.467937 L 98.822469 70.796062 L 99.392781 71.120281 L 99.959187 71.4445 L 100.5295 71.764812 L 101.095906 72.085125 L 101.666219 72.401531 L 102.232625 72.717937 L 102.802937 73.030437 L 103.369344 73.346844 L 103.939656 73.655437 L 104.506062 73.964031 L 105.076375 74.268719 L 105.642781 74.581219 L 106.213094 74.882 L 106.7795 75.182781 L 107.349812 75.479656 L 107.916219 75.784344 L 108.486531 76.081219 L 109.052937 76.378094 L 109.62325 76.671062 L 110.189656 76.967937 L 110.759969 77.253094 L 111.326375 77.542156 L 111.896687 77.827312 L 112.463094 78.124187 L 113.033406 78.41325 L 113.599812 78.698406 L 114.170125 78.983562 L 114.736531 79.264812 L 115.306844 79.542156 L 115.87325 79.823406 L 116.443562 80.10075 L 117.009969 80.378094 L 117.580281 80.651531 L 118.146687 80.924969 L 118.717 81.1945 L 119.283406 81.467937 L 119.853719 81.737469 L 120.420125 82.003094 L 120.990437 82.268719 L 121.556844 82.53825 L 122.127156 82.799969 L 122.693562 83.065594 L 123.263875 83.327312 L 123.830281 83.589031 L 124.400594 83.846844 L 124.967 84.104656 L 125.537312 84.362469 L 126.103719 84.620281 L 126.674031 84.874187 L 127.240437 85.128094 L 127.81075 85.378094 L 128.377156 85.632 L 128.947469 85.882 L 129.513875 86.132 L 130.084187 86.378094 L 130.650594 86.628094 L 131.220906 86.874187 L 131.787312 87.120281 L 132.357625 87.362469 L 132.924031 87.604656 L 133.494344 87.846844 L 134.06075 88.089031 L 134.631062 88.327312 L 135.197469 88.565594 L 135.767781 88.803875 L 136.334187 89.042156 L 136.9045 89.276531 L 137.470906 89.514812 L 138.041219 89.745281 L 138.607625 89.979656 L 139.177937 90.210125 L 139.744344 90.4445 L 140.314656 90.671062 L 140.881062 90.901531 L 141.451375 91.128094 L 142.017781 91.358562 L 142.584187 91.581219 L 143.1545 91.807781 L 143.720906 92.034344 L 144.291219 92.257 L 144.857625 92.479656 L 145.427937 92.702312 L 145.994344 92.924969 L 146.564656 93.143719 L 147.131062 93.362469 L 147.701375 93.581219 L 148.267781 93.799969 L 148.838094 94.018719 L 149.4045 94.233562 L 149.974812 94.448406 L 150.541219 94.659344 L 151.111531 94.874187 L 151.677937 95.089031 L 152.24825 95.299969 L 152.814656 95.510906 L 153.384969 95.721844 L 153.951375 95.932781 L 154.521687 96.139812 L 155.088094 96.346844 L 155.658406 96.553875 L 156.224812 96.760906 L 156.795125 96.967937 L 157.361531 97.171062 L 157.931844 97.378094 L 158.49825 97.581219 L 159.068562 97.784344 L 159.634969 97.987469 L 160.205281 98.186687 L 160.771687 98.385906 L 161.342 98.589031 L 161.908406 98.78825 L 162.478719 98.987469 L 163.045125 99.182781 L 163.615437 99.382 L 164.181844 99.577312 L 164.752156 99.772625 L 165.318562 99.967937 L 165.888875 100.16325 L 166.455281 100.354656 L 167.025594 100.549969 L 167.592 100.745281 L 168.162312 100.936687 L 168.728719 101.128094 L 169.299031 101.315594 L 169.865437 101.507 L 170.43575 101.6945 L 171.002156 101.885906 L 171.572469 102.073406 L 172.138875 102.260906 L 172.709187 102.448406 L 173.275594 102.632 L 173.845906 102.8195 L 174.412312 103.003094 L 174.982625 103.186687 L 175.549031 103.370281 L 176.119344 103.557781 L 176.68575 103.741375 L 177.256062 103.921062 L 177.822469 104.10075 L 178.392781 104.284344 L 178.959187 104.464031 L 179.5295 104.639812 L 180.095906 104.8195 L 180.666219 105.003094 L 181.232625 105.178875 L 181.802937 105.358562 L 182.369344 105.530437 L 182.939656 105.710125 L 183.506062 105.885906 L 184.076375 106.061687 L 184.642781 106.233562 L 185.213094 106.409344 L 185.7795 106.585125 L 186.349812 106.757 L 186.916219 106.928875 L 187.486531 107.10075 L 188.052937 107.272625 L 188.62325 107.440594 L 189.189656 107.612469 L 189.759969 107.784344 L 190.326375 107.952312 L 190.896687 108.124187 L 191.463094 108.292156 L 192.033406 108.460125 L 192.599812 108.624187 L 193.170125 108.792156 L 193.736531 108.960125 L 194.306844 109.128094 L 194.87325 109.292156 L 195.443562 109.456219 L 196.009969 109.624187 L 196.580281 109.784344 L 197.146687 109.948406 L 197.717 110.112469 L 198.283406 110.272625 L 198.853719 110.436687 L 199.420125 110.60075 L 199.990437 110.764812 L 200.556844 110.921062 L 201.127156 111.081219 L 201.693562 111.241375 L 202.263875 111.397625 L 202.830281 111.557781 L 203.400594 111.717937 L 203.967 111.878094 L 204.537312 112.034344 L 205.103719 112.1945 L 205.674031 112.346844 L 206.240437 112.503094 L 206.81075 112.659344 L 207.377156 112.815594 L 207.947469 112.97575 L 208.513875 113.128094 L 209.084187 113.276531 L 209.650594 113.432781 L 210.220906 113.585125 L 210.787312 113.737469 L 211.357625 113.889812 L 211.924031 114.042156 L 212.494344 114.202312 L 213.06075 114.35075 L 213.631062 114.499187 L 214.197469 114.647625 L 214.767781 114.799969 L 215.334187 114.948406 L 215.9045 115.096844 L 216.470906 115.245281 L 217.041219 115.397625 L 217.607625 115.546062 L 218.177937 115.690594 L 218.744344 115.842937 L 219.314656 115.987469 L 219.881062 116.132 L 220.451375 116.276531 L 221.017781 116.421062 L 221.588094 116.573406 L 222.1545 116.717937 L 222.724812 116.862469 L 223.291219 117.003094 L 223.861531 117.147625 L 224.427937 117.292156 L 224.99825 117.432781 L 225.564656 117.573406 L 226.134969 117.72575 L 226.701375 117.866375 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,100%);fill-opacity:0.3;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:0.3;stroke-miterlimit:10;" d="M 28.345906 -0.0008125 L 28.345906 43.835125 L 94.486531 43.835125 L 94.486531 -0.0008125 Z M 28.345906 -0.0008125 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<path style="fill:none;stroke-width:0.79701;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 28.345906 -0.0008125 L 28.345906 43.835125 L 94.486531 43.835125 L 94.486531 -0.0008125 Z M 28.345906 -0.0008125 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,100%);fill-opacity:0.3;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:0.3;stroke-miterlimit:10;" d="M 94.486531 -0.0008125 L 94.486531 85.272625 L 160.631062 85.272625 L 160.631062 -0.0008125 Z M 94.486531 -0.0008125 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<path style="fill:none;stroke-width:0.79701;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 94.486531 -0.0008125 L 94.486531 85.272625 L 160.631062 85.272625 L 160.631062 -0.0008125 Z M 94.486531 -0.0008125 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,100%);fill-opacity:0.3;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:0.3;stroke-miterlimit:10;" d="M 160.631062 -0.0008125 L 160.631062 108.952312 L 226.775594 108.952312 L 226.775594 -0.0008125 Z M 160.631062 -0.0008125 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
<path style="fill:none;stroke-width:0.79701;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 160.631062 -0.0008125 L 160.631062 108.952312 L 226.775594 108.952312 L 226.775594 -0.0008125 Z M 160.631062 -0.0008125 " transform="matrix(1,0,0,-1,40.283,194.882)"/>
</g>
</svg>
</center>
The sum is <img class="equation_image" title=" \displaystyle \sum_{i=0}^{2} \frac{7 \ln{\left(\frac{7 i}{3} + \frac{13}{6} \right)}}{3} = 9.7979 " src="/equation_images/%20%5Cdisplaystyle%20%5Csum_%7Bi%3D0%7D%5E%7B2%7D%20%5Cfrac%7B7%20%5Cln%7B%5Cleft%28%5Cfrac%7B7%20i%7D%7B3%7D%20%2B%20%5Cfrac%7B13%7D%7B6%7D%20%5Cright%29%7D%7D%7B3%7D%20%3D%209.7979%20" alt="LaTeX: \displaystyle \sum_{i=0}^{2} \frac{7 \ln{\left(\frac{7 i}{3} + \frac{13}{6} \right)}}{3} = 9.7979 " data-equation-content=" \displaystyle \sum_{i=0}^{2} \frac{7 \ln{\left(\frac{7 i}{3} + \frac{13}{6} \right)}}{3} = 9.7979 " /> . </p> </p>