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Use 3 rectangles to approximate \(\displaystyle \int\limits_{1}^{7} \ln{\left(x \right)}\, dx\) using the midpoint. Draw a graph showing the sum.
\(\displaystyle \Delta x = \frac{b-a}{n}=2\) and \(\displaystyle x_i = a+i\Delta x +\frac{\Delta x}{2} = 2 i + 2\). Drawing a graph gives:
\begin{tikzpicture}[yscale=2]
\draw[very thick, color=blue, opacity=0.3, xstep=1, ystep=1] (-1,-1) grid (8,2.10000000000000);
\draw[latex-latex, very thick] (-1.4,0)--(8.4,0) node[right]{$x$};
\draw[latex-latex, very thick] (0,-1.1)--(0,2.2) node[above]{$y$};
\foreach \x in { -1,1,...,8 }{
\pgfmathsetmacro{\xtick}{\x != 0 ? "\x" : ""}
\draw (\x, .2) -- (\x,-.2) node[below]{$\xtick$};
}
\foreach \y in { -1,-0.5,...,2.10000000000000 }{
\pgfmathsetmacro{\ytick}{\y != 0 ? "\y" : ""}
\draw (.2, \y) -- (-.2, \y) node[left]{$\ytick$};
}
\draw[thick, color=blue, domain={1.0}:{7.0}, samples=200, samples =350] plot(\x, {ln((\x))});
\draw[fill=blue, opacity=0.3](1.0000, 0) rectangle (3.0000, 0.69315);
\draw[thick](1.0000, 0) rectangle (3.0000, 0.69315);
\draw[fill=blue, opacity=0.3](3.0000, 0) rectangle (5.0000, 1.3863);
\draw[thick](3.0000, 0) rectangle (5.0000, 1.3863);
\draw[fill=blue, opacity=0.3](5.0000, 0) rectangle (7.0000, 1.7918);
\draw[thick](5.0000, 0) rectangle (7.0000, 1.7918);
\end{tikzpicture}
\begin{question}Use 3 rectangles to approximate $\int\limits_{1}^{7} \ln{\left(x \right)}\, dx$ using the midpoint. Draw a graph showing the sum.
\soln{9cm}{$\Delta x = \frac{b-a}{n}=2$ and $x_i = a+i\Delta x +\frac{\Delta x}{2} = 2 i + 2$. 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 (8,2.10000000000000);
\draw[latex-latex, very thick] (-1.4,0)--(8.4,0) node[right]{$x$};
\draw[latex-latex, very thick] (0,-1.1)--(0,2.2) node[above]{$y$};
\foreach \x in { -1,1,...,8 }{
\pgfmathsetmacro{\xtick}{\x != 0 ? "\x" : ""}
\draw (\x, .2) -- (\x,-.2) node[below]{$\xtick$};
}
\foreach \y in { -1,-0.5,...,2.10000000000000 }{
\pgfmathsetmacro{\ytick}{\y != 0 ? "\y" : ""}
\draw (.2, \y) -- (-.2, \y) node[left]{$\ytick$};
}
\draw[thick, color=blue, domain={1.0}:{7.0}, samples=200, samples =350] plot(\x, {ln((\x))});
\draw[fill=blue, opacity=0.3](1.0000, 0) rectangle (3.0000, 0.69315);
\draw[thick](1.0000, 0) rectangle (3.0000, 0.69315);
\draw[fill=blue, opacity=0.3](3.0000, 0) rectangle (5.0000, 1.3863);
\draw[thick](3.0000, 0) rectangle (5.0000, 1.3863);
\draw[fill=blue, opacity=0.3](5.0000, 0) rectangle (7.0000, 1.7918);
\draw[thick](5.0000, 0) rectangle (7.0000, 1.7918);
\end{tikzpicture}
\end{center}
The sum is $\sum_{i=0}^{2} 2 \ln{\left(2 i + 2 \right)} = 7.7424$. }
\end{question}
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\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}^{7} \ln{\left(x \right)}\, dx " src="/equation_images/%20%5Cdisplaystyle%20%5Cint%5Climits_%7B1%7D%5E%7B7%7D%20%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%5C%2C%20dx%20" alt="LaTeX: \displaystyle \int\limits_{1}^{7} \ln{\left(x \right)}\, dx " data-equation-content=" \displaystyle \int\limits_{1}^{7} \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}=2 " src="/equation_images/%20%5Cdisplaystyle%20%5CDelta%20x%20%3D%20%5Cfrac%7Bb-a%7D%7Bn%7D%3D2%20" alt="LaTeX: \displaystyle \Delta x = \frac{b-a}{n}=2 " data-equation-content=" \displaystyle \Delta x = \frac{b-a}{n}=2 " /> and <img class="equation_image" title=" \displaystyle x_i = a+i\Delta x +\frac{\Delta x}{2} = 2 i + 2 " src="/equation_images/%20%5Cdisplaystyle%20x_i%20%3D%20a%2Bi%5CDelta%20x%20%2B%5Cfrac%7B%5CDelta%20x%7D%7B2%7D%20%3D%202%20i%20%2B%202%20" alt="LaTeX: \displaystyle x_i = a+i\Delta x +\frac{\Delta x}{2} = 2 i + 2 " data-equation-content=" \displaystyle x_i = a+i\Delta x +\frac{\Delta x}{2} = 2 i + 2 " /> . Drawing a graph gives:
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The sum is <img class="equation_image" title=" \displaystyle \sum_{i=0}^{2} 2 \ln{\left(2 i + 2 \right)} = 7.7424 " src="/equation_images/%20%5Cdisplaystyle%20%5Csum_%7Bi%3D0%7D%5E%7B2%7D%202%20%5Cln%7B%5Cleft%282%20i%20%2B%202%20%5Cright%29%7D%20%3D%207.7424%20" alt="LaTeX: \displaystyle \sum_{i=0}^{2} 2 \ln{\left(2 i + 2 \right)} = 7.7424 " data-equation-content=" \displaystyle \sum_{i=0}^{2} 2 \ln{\left(2 i + 2 \right)} = 7.7424 " /> . </p> </p>