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\centerline{MATH 134 -- Calculus with Fundamentals 2}
\centerline{Practice Day on the $L_N,R_N,M_N$ Riemann Sums}
\centerline{January 25, 2018}
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\noindent
{\it Background}
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We have now introduced the $L_N,R_N,M_N$ sum approximations to the 
area between the graph $y = f(x)$ and the $x$-axis, for $x$ in $[a,b]$.
All three sums are based on a \emph{partition}, or subdividision, of the interval $[a,b]$ into some
given number $N$ of smaller intervals of width $\Delta x = \frac{b - a}{N}$.  The
endpoints of the smaller intervals are then given by 
$$x_j = a + j\Delta x, \quad {\rm for} \quad j = 0,1,2,\ldots, N.$$
The midpoint of the $j$th subinterval is 
$$m_j = \frac{x_{j-1} + x_j}{2}.$$
Then we have 
\begin{itemize}
\item using the left endpoints:  $L_N = \sum_{j = 1}^N f(x_{j-1}) \Delta x$ 
\item using the right endpoints: $R_N = \sum_{j = 1}^N f(x_j) \Delta x$
\item using the midpoints: $M_N = \sum_{j = 1}^N f(m_j) \Delta x =  \sum_{j = 1}^N f\left(\frac{x_{j-1} + x_j}{2}\right) \Delta x$.  
\end{itemize}
These are all special cases of the more general \emph{Riemann sums}, named for the German 
Georg Friedrich Bernhard Riemann (1826 - 1866), who was one of the most celebrated and original
mathematicians of the 19th century.  
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\noindent
{\it Questions}
\begin{enumerate}
\item[(A)]  Let $f(x) = x^2 + 3x + 1$ on the interval $[a,b] = [2,4]$.  
\begin{enumerate}
\item[(1)] Take $N = 3$ and compute
the $L_3,R_3,M_3$ sums for this function.
\item[(2)]  With the same function and the same interval, now take $N = 6$
and compute the $L_6, R_6, M_6$ sums for this function.
\item[(3)]  Which is closer to the true area under the graph $y = x^2 + 3x + 1$ on $[2,4]$, 
$L_3$ or $L_6$?  Explain by drawing the picture of the graph and the approximating 
rectangles.
\item[(4)]  Similarly, which is closer to the true area under the graph, $R_3$ or $R_6$?
\item[(5)]  Of the six numbers you computed in parts (1) and (2), which do you think
is \emph{the closest} to the true area under the graph?  Explain.
\end{enumerate}
\item[(B)]  Repeat parts (1) and (2) of question A for $f(x) = \sin(x)$ on the interval $[a,b] = [0,\pi]$.  
Explain why the answers to parts (3), (4), and (5) are not as easy to see in this case(!)
\end{enumerate}
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