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\centerline{MATH 135 -- Calculus 1}
\centerline{Practice on Approximating Areas}
\centerline{December 2, 2019}
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\noindent
{\it Background}
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We have discussed the idea of approximating the area between a graph
$y = f(x) \ge 0$ and the $x$-axis for $a \le b\le x$ by using sums
of areas of rectangles and worked out a first example.  To practice
using these ideas you will work through another similar example using
a different function, interval, and subdivision.  Here's the description
of the region we want in formulas:  The region between $y = x^2 - 4x + 6$
and the $x$-axis, for $1 \le x \le 2$.  
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\noindent
{\it Questions}
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\begin{enumerate}
\item Sketch the region (you may want to complete the square in $x^2 - 4x + 6$
to do this).  Check your work by looking at the figure on the back of this page.
\item To subdivide the $x$-interval $[1,2]$ into $N = 5$ equal smaller intervals, what 
$\Delta x$ should we use?  What are the endpoints $x_0,x_1,x_2,x_3,x_4,x_5$
of the intervals?
\item Make rectangles using the value of the function at the \emph{left endpoint}
of each smaller interval to get the heights. 
Sketch these rectangles together with the graph.
\item Write the sum giving the areas
of these rectangles using summation notation, and compute the numerical value.
\item Now make rectangles using the value of the function at the \emph{right endpoint}
of each smaller interval to get the heights.  Sketch these rectangles together with the graph.
\item  Write the sum giving the areas
of these rectangles using summation notation, and compute the numerical value.
\item What can you say about the area of the region from your computations?  
\end{enumerate}
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\begin{figure}
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\caption{The region}
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