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\title{Math 242, Lecture 8}
\maketitle

\subsection{ Practice with the definite integral and the Fundamental Theorem} 

\begin{example}
Use both the Fundmental Theorem and a  Riemann sum with 50 terms to evaluate
$\int_1^3 \frac{x}{x+1}$.  
\end{example}


\section{ Further applications of the definite integral}  


Many quantities can be well-approximated  
(to {\em first order})   by a Riemann sum 
$f(x_1) \Delta x + f(x_2) \Delta x + \cdots + f(x_n) \Delta x$  
Any such
quantity can be computed with a  definite integral.  
The FTC can  be applied to compute the integral,    
so the FTC is useful well beyond computing areas under
or between curves.  

The steps for setting up integrals to compute quantities
in many different contexts are as follows:  
\begin{itemize}
\item Name a variable through which the quantity 
can naturally be ``broken up''  
(for the sake of concreteness, let us name that variable $x$).  
\item For small ranges of values for that variable, estimate
the value of that quantity.   Your answer should be
of the form $f(x) \cdot \Delta x$,   where $\Delta x$ represents
the change in the variable over that range   and $f(x)$
is the function which makes the estimate work.  
\item Determine the range of values  the variable must
take on to account for the entire quantity  
(for the sake of concreteness, call those $a$ and $b$).  
\item Evaluate $\int_a^b f(x) dx$ in order to compute 
the quantity ! 
\end{itemize}

We begin with examples coming from mathematics and
physics,   as we often like to do for first illustrations, 
since it is in these subjects which the calculus was developed.  
We will not emphasize these on homework and tests,   but they
are important because they beautifully illustrate the essence
of modeling with the definite integral.  

\begin{example}
Find the volume of the football obtained by rotating 
the curve $y = \frac{1}{2} (1 - x^2)$ between $x = -1$ and $x = 1$
around the $x$-axis in three-dimensional space.  
To be concrete, what is the volume of a foot-long football?  
\begin{itemize}
\item variable: x   we will ``slice'' the football.  
\item begin by estimating the volume near $x = 0$ with $\Delta x = 0.1$  
using (area of base) times (height).   namely $\pi (\frac{1}{2})^2 \times  0.1$.
in general we get $\pi \left(\frac{1}{2}(1 - x^2)\right)^2 \Delta x$.
\item $x$ ranges from $-1$ to $1$.  
\item Evaluate $\int_{-1}^1 \pi \left(\frac{1}{2}(1 - x^2)\right)^2 dx$.  
\end{itemize}
\end{example}

Using similar techniques one can compute the volume of spheres,
cones, donuts, and satellite dishes.  

These kinds of applied integration problems often comprise
an entire course!    In fact, the technique of chopping a 
problem into (tiny) pieces,   analyzing the pieces,  
and then {\em integrating} the pieces into a whole is what
gave the subject its name.  

\begin{example}
A metal rod is made with an alloy of two metals,
metal A which weighs $2$ grams per $cm$ of length and 
metal B which weighs $5$ grams per $cm$ of length.   
Suppose that one end of a $10 cm$ rod is all metal B and
the other end is a fifty-fifty mix, and the mix of metals
changes linearly.    How much does the rod weigh?  
\end{example}

\section{ Areas between curves}

We can use this same Riemann sum analysis to quickly
evaluate areas between two curves.

To find the area between the graph of $f(x)$
and that of $g(x)$ between $x = a$ and $x = b$ 
   we evaluate $\int_a^b | f(x) - g(x) | dx$. \\  

\begin{example} 
Find the area of the region between the functions $f(x) = x$ and
$g(x) = x^2$ and the vertical lines at $x = -3$ and $x = -1$.
\end{example}

Sometimes there are no vertical lines supplied; one
must figure out where the curves intersect.  

\begin{example}
Find the total area of the football-shaped region
between the graphs of $f(x) = x^2 - 2$  and $g(x) = 2 - x^2$.  
\end{example}


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