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Use 5 rectangles to approximate \(\displaystyle \int\limits_{5}^{9} \frac{4}{x}\, dx\) using the midpoint. Draw a graph showing the sum.
\(\displaystyle \Delta x = \frac{b-a}{n}=\frac{4}{5}\) and \(\displaystyle x_i = a+i\Delta x +\frac{\Delta x}{2} = \frac{4 i}{5} + \frac{27}{5}\). Drawing a graph gives:
\begin{tikzpicture}[yscale=5]
\draw[very thick, color=blue, opacity=0.3, xstep=1, ystep=1] (-1,-1) grid (10,1.05);
\draw[latex-latex, very thick] (-1.4,0)--(10.4,0) node[right]{$x$};
\draw[latex-latex, very thick] (0,-1.1)--(0,1.1) node[above]{$y$};
\foreach \x in { -1,1,...,10 }{
\pgfmathsetmacro{\xtick}{\x != 0 ? "\x" : ""}
\draw (\x, .2) -- (\x,-.2) node[below]{$\xtick$};
}
\foreach \y in { -1,-0.75,...,1.05 }{
\pgfmathsetmacro{\ytick}{\y != 0 ? "\y" : ""}
\draw (.2, \y) -- (-.2, \y) node[left]{$\ytick$};
}
\draw[thick, color=blue, domain={5.0}:{9.0}, samples=200, samples =350] plot(\x, {4/(\x)});
\draw[fill=blue, opacity=0.3](5.0000, 0) rectangle (5.8000, 0.74074);
\draw[thick](5.0000, 0) rectangle (5.8000, 0.74074);
\draw[fill=blue, opacity=0.3](5.8000, 0) rectangle (6.6000, 0.64516);
\draw[thick](5.8000, 0) rectangle (6.6000, 0.64516);
\draw[fill=blue, opacity=0.3](6.6000, 0) rectangle (7.4000, 0.57143);
\draw[thick](6.6000, 0) rectangle (7.4000, 0.57143);
\draw[fill=blue, opacity=0.3](7.4000, 0) rectangle (8.2000, 0.51282);
\draw[thick](7.4000, 0) rectangle (8.2000, 0.51282);
\draw[fill=blue, opacity=0.3](8.2000, 0) rectangle (9.0000, 0.46512);
\draw[thick](8.2000, 0) rectangle (9.0000, 0.46512);
\end{tikzpicture}
\begin{question}Use 5 rectangles to approximate $\int\limits_{5}^{9} \frac{4}{x}\, dx$ using the midpoint. Draw a graph showing the sum.
\soln{9cm}{$\Delta x = \frac{b-a}{n}=\frac{4}{5}$ and $x_i = a+i\Delta x +\frac{\Delta x}{2} = \frac{4 i}{5} + \frac{27}{5}$. Drawing a graph gives:
\begin{center}\begin{tikzpicture}[yscale=5]
\draw[very thick, color=blue, opacity=0.3, xstep=1, ystep=1] (-1,-1) grid (10,1.05);
\draw[latex-latex, very thick] (-1.4,0)--(10.4,0) node[right]{$x$};
\draw[latex-latex, very thick] (0,-1.1)--(0,1.1) node[above]{$y$};
\foreach \x in { -1,1,...,10 }{
\pgfmathsetmacro{\xtick}{\x != 0 ? "\x" : ""}
\draw (\x, .2) -- (\x,-.2) node[below]{$\xtick$};
}
\foreach \y in { -1,-0.75,...,1.05 }{
\pgfmathsetmacro{\ytick}{\y != 0 ? "\y" : ""}
\draw (.2, \y) -- (-.2, \y) node[left]{$\ytick$};
}
\draw[thick, color=blue, domain={5.0}:{9.0}, samples=200, samples =350] plot(\x, {4/(\x)});
\draw[fill=blue, opacity=0.3](5.0000, 0) rectangle (5.8000, 0.74074);
\draw[thick](5.0000, 0) rectangle (5.8000, 0.74074);
\draw[fill=blue, opacity=0.3](5.8000, 0) rectangle (6.6000, 0.64516);
\draw[thick](5.8000, 0) rectangle (6.6000, 0.64516);
\draw[fill=blue, opacity=0.3](6.6000, 0) rectangle (7.4000, 0.57143);
\draw[thick](6.6000, 0) rectangle (7.4000, 0.57143);
\draw[fill=blue, opacity=0.3](7.4000, 0) rectangle (8.2000, 0.51282);
\draw[thick](7.4000, 0) rectangle (8.2000, 0.51282);
\draw[fill=blue, opacity=0.3](8.2000, 0) rectangle (9.0000, 0.46512);
\draw[thick](8.2000, 0) rectangle (9.0000, 0.46512);
\end{tikzpicture}
\end{center}
The sum is $\sum_{i=0}^{4} \frac{16}{5 \left(\frac{4 i}{5} + \frac{27}{5}\right)} = 2.3482$. }
\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 5 rectangles to approximate <img class="equation_image" title=" \displaystyle \int\limits_{5}^{9} \frac{4}{x}\, dx " src="/equation_images/%20%5Cdisplaystyle%20%5Cint%5Climits_%7B5%7D%5E%7B9%7D%20%5Cfrac%7B4%7D%7Bx%7D%5C%2C%20dx%20" alt="LaTeX: \displaystyle \int\limits_{5}^{9} \frac{4}{x}\, dx " data-equation-content=" \displaystyle \int\limits_{5}^{9} \frac{4}{x}\, 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{4}{5} " src="/equation_images/%20%5Cdisplaystyle%20%5CDelta%20x%20%3D%20%5Cfrac%7Bb-a%7D%7Bn%7D%3D%5Cfrac%7B4%7D%7B5%7D%20" alt="LaTeX: \displaystyle \Delta x = \frac{b-a}{n}=\frac{4}{5} " data-equation-content=" \displaystyle \Delta x = \frac{b-a}{n}=\frac{4}{5} " /> and <img class="equation_image" title=" \displaystyle x_i = a+i\Delta x +\frac{\Delta x}{2} = \frac{4 i}{5} + \frac{27}{5} " 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%7B4%20i%7D%7B5%7D%20%2B%20%5Cfrac%7B27%7D%7B5%7D%20" alt="LaTeX: \displaystyle x_i = a+i\Delta x +\frac{\Delta x}{2} = \frac{4 i}{5} + \frac{27}{5} " data-equation-content=" \displaystyle x_i = a+i\Delta x +\frac{\Delta x}{2} = \frac{4 i}{5} + \frac{27}{5} " /> . Drawing a graph gives:
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The sum is <img class="equation_image" title=" \displaystyle \sum_{i=0}^{4} \frac{16}{5 \left(\frac{4 i}{5} + \frac{27}{5}\right)} = 2.3482 " src="/equation_images/%20%5Cdisplaystyle%20%5Csum_%7Bi%3D0%7D%5E%7B4%7D%20%5Cfrac%7B16%7D%7B5%20%5Cleft%28%5Cfrac%7B4%20i%7D%7B5%7D%20%2B%20%5Cfrac%7B27%7D%7B5%7D%5Cright%29%7D%20%3D%202.3482%20" alt="LaTeX: \displaystyle \sum_{i=0}^{4} \frac{16}{5 \left(\frac{4 i}{5} + \frac{27}{5}\right)} = 2.3482 " data-equation-content=" \displaystyle \sum_{i=0}^{4} \frac{16}{5 \left(\frac{4 i}{5} + \frac{27}{5}\right)} = 2.3482 " /> . </p> </p>