\def\wsn{23}
\input worksheet.tex

\noindent
{\bf Note:} In this worksheet, you will be expected to work with {\it
Riemann sums} (sometimes called {\it summation notation}).  If
$a_1,a_2,a_3,\ldots,a_i,\ldots,a_{n-1},a_n$ are $n$ numbers, then the
meaning of the Riemann sum $\displaystyle{\sum_{i=1}^na_i}$ is
$$
a_1+a_2+a_3+\cdots+a_i+\cdots+a_{n-1}+a_n.
$$

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\item{1.}  Consider the following definitions:

\medskip
\item{} {\bf Definition 1.}  Let $a=x_0\leq x_1\leq x_2\leq\cdots\leq
x_n=b$ be a partition of the interval $[a,b]$.  The {\it mesh} of the
partition is the width of the largest subinterval $[x_{i-1},x_i]$.

\medskip
\item{} {\bf Definition 2.}  If $f$ is a function defined on $[a,b]$
and the sums $\sum_{i=1}^nf(c_i)(x_i-x_{i-1})$ approach a certain
number as the mesh of partitions of $[a,b]$ shrinks toward 0 (no
matter how the sampling numbers $c_i$ are chosen in $[x_{i-1},x_i]$),
that certain number is called {\it the definite integral of $f$ from
$a$ to $b$.}  It is denoted
$$
\int_a^bf(x)\,dx.
$$
\medskip
\item{}
In short, the definite integral of $f$ over $[a,b]$ is
$$
\lim_{{\rm mesh}\to0}\sum_{i=1}^nf(c_i)\Delta x_i.
$$

\itemitem{a)}  Draw a picture of a sequence of partitions in which the
mesh does not go to 0.  

\medskip
\itemitem{b)}  Explain why a sequence of partitions such as the ones
you drew in part a) are not be allowed in Definition 2.  

\medskip
\itemitem{c)}  Sketch two identical copies of the graph of some
function $f$ on an interval $[a,b]$.  Partition the interval in both
pictures, using a finer partition for the second graph.  Now, draw in
rectangles which explain the definition of the definite integral.

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\item{2.}  Write the following limits as definite integrals:
$$
\lim_{n\to\infty}\sum_{i=1}^n f\left(a+i{b-a\over n}\right){b-a\over n}
\hskip.5in
\lim_{n\to\infty}\sum_{i=1}^n \left(2+{2i\over n}\right)^2{2\over n}
$$

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\item{3.}  Write the following definite integrals as limits:
$$
\int_0^1x+2\,dx\hskip.5in\int_{-3}^3x^2-x\,dx
$$

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\item{4.}  Use the formulas
$$
\sum_{i=1}^n1=n\hskip.5in
\sum_{i=1}^ni={n(n+1)\over2}\hskip.5in
\sum_{i=1}^ni^2={n(n+1)(2n+1)\over6}
$$
to evaluate the Riemann sums you wrote down in Problem 3.  Then
compute the limits as $n\to\infty$.

\bigskip
\item{5.}  Use the Second Fundamental Theorem of Calculus to check your
answers in Problem 4.




\bye