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\title{Math 1a.  Lecture 8 \\
More about Areas}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To  be able to calculate the area between two curves and to understand the connection between 
$$
\int_a^b f(x) - g(x) \, dx
$$
and Riemann Sums.
 
\end{itemize} 



\section{Finding the Area between Two Curves}

Suppose that we wish to find the area between $\sin x$ and $\cos x$ on the interval $[\pi/4, 5\pi/4]$.


Let $f(x)$ and $g(x)$ be defined on an interval $[a, b]$ with $f(x) \geq g(x)$ for all $x \in [a, b]$.  If we wish to approximate the area between the two curves, we can partition the interval as
$$
a = x_0 < x_1 < x_2 < \cdots < x_{n-1} < x_n = b
$$
and estimate the area as a Riemann Sum,
$$
\text{Area} = \sum_{i=1}^n A_i \approx \sum_{i=1}^n [f(x_i^*) - g(x_i^*)] \Delta x,
$$
where $\Delta x = x_i - x_{i - 1}$ and $x_i^* \in [x_{i-1}, x_i]$.

Thus, we can define the area between two curves to be a Riemann integral
$$
\text{Area} = \int_a^b   [f(x) - g(x)] \,  dx.
$$

Thus, the area between $\sin x$ and $\cos x$ on the interval $[\pi/4, 5\pi/4]$ is
$$
\int_{\pi/4}^{5 \pi/4} \sin x - \cos x \, dx = -\cos x - \sin x \Big|_{\pi/4}^{5 \pi/4} = 2 \sqrt{2}.
$$

\section{Worksheet}

\begin{enumerate}

\item 
Find the area bounded by
\begin{eqnarray*}
x & = & 0 \\
x & = & 2 \\
y & = & \frac{1}{x+1} \\
y & = & \sqrt{x + 2}
\end{eqnarray*}


\item 
Find the area bounded by
\begin{eqnarray*}
y & = & x \\
y & = & x^2
\end{eqnarray*}



\item 
Find the area bounded by
\begin{eqnarray*}
x & = & y^2 - 4y \\
x & = & 2y - y^2 
\end{eqnarray*}



\item
A cross-section of an airplane wing has length 200 cm.  Measurements of the thickness of the wing, in centimeters, at 20-centimeter intervals are 5.8, 20.3, 26.7, 29.0, 27.6, 27.3, 23.8, 20.5, 15.1, 8.7, and 2.8.  Use Simpson's Rule to estimate the are of the wing's cross-section.



\item 
Find the area bounded by
\begin{eqnarray*}
y & = & |x| \\
y & = & x^2 - 2 \\
\end{eqnarray*}


\item 
Find the area bounded by
\begin{eqnarray*}
y & = & 1/x \\
y & = & x \\
y & = & x/4
\end{eqnarray*}

\end{enumerate}




\section*{References}

\begin{itemize}

\item
\S 6.1 in 
James Stewart.
{\it Single Variable Calculus: Concepts \& Context}, third edition.
Brooks/Cole, Belmont CA, 2005.
ISBN 0--534--41022--7.


\end{itemize}



\section*{Notes}

\subsection*{February 27, 2006}



\end{document}

