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\begin{center} \textbf{The R-integral} \end{center}

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Definition.  Let $[a,b]$ be an interval.  Then $S$ is a subdivision of $[a,b]$ means that $S$ is a finite increasing sequence $\{x_i\}_{i=0}^n$ so that $a=x_0 < x_1 < x_2 < \dots < x_{n-1} < x_n = b$.

Definition.  If $S = \{x_i\}_{i=0}^n$ is a subdivision of the interval $[a,b]$ then $\mbox{mesh}(S) = \sup\{(x_i - x_{i-1}) \ | \ 0<i\le n \}$.

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Definition.  Suppose that $[a,b]$ is an interval and $f: [a,b] \rightarrow \mathbb{R}$ is a function.  Then $f$ is integrable over the interval $[a,b]$ means that there is a number $I$ so that if $\epsilon > 0$ is a positive number, then there exists a positive number $\delta > 0$ so if $S = \{x_i\}_{i=1}^n$ is a subdivision of $[a,b]$ with $\mbox{mesh}(S)<\delta$ and for each $i$, $x_i^* \in [x_{i-1}, x_i]$ then
$$\Big{|}\sum_{i=1}^n f(x_i^*)(x_i - x_{i-1}) - I \Big{|} < \epsilon.$$
The number $I$ is called the integral of the function $f$ over the interval $[a,b]$ and it is denoted by
$$\int_a^b f(x) dx \mbox{ or } \int_{[a,b]}f.$$
We will use both notations.

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Exercise 2.0.  (a.) Show that the number $I$ in the definition of the integral is unique.

(b.) Show that if $f$ is integrable on $[a,c]$ and $a<b<c$ then $f$ is integrable on $[a,b]$.

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Theorem 2.1.  If $f:[a,b] \rightarrow \mathbb{R}$ is a continuous function, then it is integrable over the interval $[a,b]$.

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Exercise 2.1.  In the following exercises the function $f$ is defined for a particular interval $[a,b]$.  Determine from the definition if $\int_{[a,b]}f$ exists and if it does, determine its value.

$$\begin{array}{lll}
a.) & f(x) = 1,  & x \in [a,b]. \\
b.) & f(x) = x,  & x \in [0,1]. \\
c.) & f(x) = x^2,  & x \in [0,1]. \\
  \\
d.) & f = \left \{
\begin{array}{lll}
1 & \mbox{if } 0 \le x < 1 \\
3 & \mbox{if } 1 \le x \le 2
\end{array} \right . & x \in [0,2]. \\
  \\
e.) & f = \left \{
\begin{array}{lll}
0 & \mbox{if } x=0 \\
\frac 1x & \mbox{if } 0< x \le 1
\end{array} \right . & x \in [0,1]. \\
  \\
f.) & f = \left \{
\begin{array}{lll}
0 & \mbox{if } x \mbox{ is rational} \\
1 & \mbox{if } x \mbox{ is irrational}
\end{array} \right . & x \in [0,1].
\end{array} $$

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Exercise 2.2.  If $f$ is integrable over the interval $[a,b]$ then:
$$\int_{[a,b]}f = \lim_{n \rightarrow \infty} \sum_{i=1}^n f\Big{(}a + i \frac{b-a}{n}\Big{)} \frac{b-a}n.$$
Is the converse true (i.e. if the limit on the right exists then is $f$ integrable)?

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Notation:  If $b<a$ then $\int_{[a,b]}f = - \int_{[b,a]}f$; $\int_a^a f(x)dx = 0$.

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Theorem 2.2.  Suppose that $f$ and $g$ are continuous functions whose domains include the intervals over which they are integrated, $c$ is the constant function and all the quantities in the following formulas are defined. Then:

\begin{eqnarray}
\int_a^b cf(x) dx & = & c \int_a^b f(x) dx \\
\int_a^b f(x)+g(x) dx & = & \int_a^b f(x) dx + \int_a^b g(x) dx \\
\int_a^b f(x) dx + \int_b^c f(x) dx & = & \int_a^c f(x) dx
\end{eqnarray}

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Theorem 2.3. Suppose that $f$ and $g$ are continuous functions whose domains include the interval $[a,b]$ and that $f(x) \le g(x)$.  Then

\qquad a. $\int_{[a,b]} f \le \int_{[a,b]}g$.

\qquad b. If there is a value $c \in [a,b]$ so that $f(c) < g(c)$ then $\int_{[a,b]} f < \int_{[a,b]} g$.

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Theorem 2.4. Suppose that $f$ is a continuous function whose domain includes the interval $[a,b]$ and that for each number $a \le x \le b$ we define $F(x) = \int_{[a,x]}f$.  Then

\qquad a. $F$ is continuous for each $x \in [a,b]$;

\qquad b. $F$ is differentiable for each $x \in [a,b]$;

\qquad c. [The fundamental theorem of calculus.] $F'(t) = f(t)$ for each $t \in [a,b]$.

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Theorem 2.5. Suppose that $f$ is a continuous function whose domain includes the interval $[a,b]$ then there exists a number $c$ with $a< c <b$ so that
$$\int_a^b f(x)dx = f(c)(b-a).$$

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Exercise 2.3.  Suppose that $f$ and $g$ are continuous functions whose domains include the interval $[a,b]$, $ a < c <b$ and the function $h$ is defined by:
$$h(x) = \left \{
\begin{array}{ll}
 f(x) & \mbox{if } a \le x < c \\
 g(x) & \mbox{if } c \le x \le b.
\end{array} \right .$$
Then $h$ is integrable over $[a,b]$ and $\int_{[a,b]}h =  \int_{[a,c]}f + \int_{[c,b]}g$.

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