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\begin{document}

\title{MTH 1420, Spring 2012}
\author{Section 5.1:  Area under a curve}
\date{}
\maketitle


\section{Introduction}

First, let us recall the big ideas related to the invention of calculus:

\vspace{2in}

\section{Approximating the area under a curve}

Actually calculating the area between a curve and the $x$-axis can be tricky, but we do have a straightforward method of \emph{approximating} it (although the calculation can be cumbersome).  Namely, we can divide our interval into smaller chunks and then make rectangles which approximate the area.  Calculating the area of rectangles is easy, so this gives us a method of approximation.

\bexa
	Approximate the area under the curve $\d y=2x$ on the interval $[0,4]$ using 2, 4 and 8 subintervals.
\eexa

\newpage


\bexa
	Find the area under the curve $\d y=2x$ on the interval $[0,4]$ by using an area formula from geometry.
\eexa
\vspace{1.5in}

\bexe \label{cubic}
	Approximate the area under the curve of $f(x)=x^3$ from $x=0$ to $x=2$, using 2, 4 and 10 subintervals.
\eexe
\vspace{3in}


\section{Calculating the area exactly}

The greater number of subintervals you use, the greater the number of rectangles, and thus the better the approximation.  If you wanted to calculate the area under a curve \emph{exactly}, you would need to use a limit.  Let $R_n$ be the approximation for the area when you use $n$ subintervals.  Thus for Exercise \ref{cubic} above we would have $R_2=\blank{.5}$,  $R_4=\blank{.5}$, and $R_{10}=\blank{.5}$.  If we let $A$ be the actual area under the curve, we have:
\newpage


We are not guaranteed that any particular limit will exist, but if $f$ is continuous, we can actually show that the limit above will \underline{always} exist. (Although we will not do it here.)

As you might imagine, calculating such a limit can be rather annoying.  This section focuses on \emph{approximating} the area under a curve, and in the next section we will actually use limits (such as above) to actually give the exact value for the area under a curve.  We will illustrate finding the \emph{exact} area now with one example.

\bexa
	Calculate the exact area under the curve of $f(x)=x^3$ from $x=0$ to $x=2$.
\eexa
\vspace{3.5in}

 \brem \textbf{Sigma notation} is used to write sums when you are adding up a bunch of terms.  We will not really need it for this section, but I will start to use it so that you can get used to it.  It looks like:
 \vspace{1in}
 \erem


\section{Evaluating at other points in a subinterval}
 
So far we have only approximated the area under the curve using equal subintervals and using the \emph{right} endpoint for calculations.  In general, however, the length of the subintervals does not need to be equal (though we will almost always use equal subintervals because it makes calculation easier), and you can evaluate the height of the rectangles at any point in the subinterval.  The two most common other places to evaluate a subinterval are at the midpoints and at the left endpoint.  

\bexa
	Calculate $M_4$ (that is, the approximation for the area under the curve using 4 subintervals and evaluating at the \emph{midpoint}) for $f(x)=x^3$ from $x=0$ to $x=2$.
\eexa 
\vspace{2in}

\bexe
	Calculate $L_4$ (that is, the approximation for the area under the curve using 4 subintervals and evaluating at the \emph{left} endpoint of each subinterval) for $f(x)=x^3$ from $x=0$ to $x=2$.
\eexe
\vspace{2in}


\section{An important example problem}

A common example of when you would need to find the area under a curve is the \underline{distance problem}.  This example is important because it illustrates a connection between the area under a curve and antiderivatives.
 
 \bexa
 	Below is the velocity graph of a braking car.  Estimate the distance traveled by the car while the brakes are applied (this is also known as the stopping distance) by finding $M_6$, that is, the area under the curve using six equal subintervals and evaluating at the \emph{midpoint}.
 \eexa






\end{document} 