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\centerline{MATH 134 -- Calculus with Fundamentals 2}
\centerline{Practice Day on Properties of $\int_a^b f(x)\ dx$}
\centerline{January 29, 2018}
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\noindent
{\it Background}
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We have now introduced the definite integral $\int_a^b f(x)\ dx$ and discussed:
\begin{itemize}
\item the interpretation of of this number as a \emph{signed area}, 
\item the \emph{interval additivity property}:  
If $f$ is integrable on $[a,b]$ and $a \le c \le b$, then 
$$\int_a^b f(x)\ dx = \int_a^c f(x)\ dx + \int_c^b f(x)\ dx$$
\item the \emph{comparison property}: If $f$ is integrable on $[a,b]$ and
$m \le f(x) \le M$ for all $x$ in $[a,b]$, then
$$m\cdot (b - a) \le \int_a^b f(x)\ dx \le M\cdot (b - a).$$  
\end{itemize}

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\noindent
{\it Questions}
Let $f(x)$ be the function on the interval $[0,6]$ plotted on the back.  Thinking
of the definite integral as \emph{signed area}, answer these questions.  
\begin{enumerate}
\item[(1)]  What are $\int_0^4 f(x)\ dx$, $\int_4^6 f(x)\ dx$, $\int_0^6 f(x)\ dx$?
\item[(2)]  From your answers to part (1) verify that 
$\int_0^6 f(x)\ dx = \int_0^4 f(x)\ dx + \int_4^6 f(x)\ dx$. 
\item[(3)]  What is $\int_5^6 f(x)\ dx$?  How can you tell?
\item[(4)]  Let $m$ be the minimum value of $f(x)$ on $[3,5]$ and
let $M$ be the maximum value of $f(x)$ on $[3,5]$ (use the information 
from the graph).  Find $m,M$ and verify that
$$m\cdot (5 - 3) \le \int_3^5 f(x)\ dx \le M\cdot (5 - 3)$$
\end{enumerate}
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\caption{The region}
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