\section{Anti-derivatives and Areas}
\subsection{ Summary of anti-derivatives}
The table below summarizes the anti-derivatives that should be memorized:
\begin{center}
\begin{tabular}{|c|c|}\hline
\textbf{$f(x)$} & \textbf{$\dps\int f(x) dx$}\rule{0pt}{16pt}\\\hline
$x^n $, for $n\not= -1$ &$\dfrac{1}{n+1}x^{n+1}+C$\rule{0pt}{16pt}\\
$\dfrac 1 x$ & $\ln(x) +C$\rule{0pt}{16pt}\\
$e^{ax}$ & $\dfrac{1}{a}e^{ax}+C$ \rule{0pt}{16pt}\\
$\cos(a x)$ & $\dfrac 1 a \sin(ax)+C$ \rule{0pt}{16pt}\\
$\sin(ax)$ & $-\dfrac 1 a \cos(ax)+C$ \rule{0pt}{16pt}\rule[-12pt]{0pt}{12pt}\\\hline
\end{tabular}
\end{center}
For any other $f(x)$ use the Mathematica command \verb+DSolve[F'[x]==f[x],F[x],x]+

\subsection{Calculating areas under curves}
\subsubsection{Theoretical summary}
We have seen that we can find the area under a curve inbetween two
values of $x$ we can sum the areas of rectangles under the curve, and take the limit as the width of the rectangles tends to zero. In
other words calculate:
\[
\lim_{dx\to 0}\sum_{i=0}^{\lfloor(b-a)/dx\rfloor} f(a+idx)\; dx
\]
In Mathematica the command for this is
\begin{verbatim}
Limit[ Sum[ f[a+i*dx], {i,0,(b-a)/dx-1} ], dx->0]
\end{verbatim}
\subsubsection{Example}
For the function $f(x)=-2x^3+8x$, calculate the following areas:
\begin{enumerate}\renewcommand{\theenumi}{\alph{enumi}}
\item The sum of the areas of rectangles under the curve of width 0.2 from $x=0.6$ to $x=1.6$. Provide a graphical illustration.
\item The sum of the areas of rectangles under the curve of width 0.05
from $x=0.6$ to $x=1.6$. Provide a graphical illustration.
\item The sum of the areas of rectangles under the curve of width 0.01
from $x=0.6$ to $x=1.6$. Provide a graphical illustration.
\item The area under the curve from $x=0.6$ to $x=1.6$.
\item The sum of the areas of rectangles under the curve of width 0.2 from $x=0.6$ to $x=3.6$. Provide a graphical illustration.
\item The sum of the areas of rectangles under the curve of width 0.01
from $x=0.6$ to $x=3.6$. Provide a graphical illustration.
\item The area under the curve from $x=0.6$ to $x=3.6$. (\textbf{Note} Areas below the $x$-axis are negative).
\end{enumerate}

\textbf{See file }\verb+ares.nb+ \textbf{for the solution.}\\

\subsubsection{Exercise for Homework 2}\label{M4_HW2} 
For the function $f(x)=\sin(x)$, calculate the following:
\begin{enumerate}\renewcommand{\theenumi}{\alph{enumi}}
\item The sum of the areas of rectangles under the curve of width 0.2 from $x=0$ to $x=\frac{\pi}{2}$. Provide a graphical illustration.
\item The sum of the areas of rectangles under the curve of width 0.05
from $x=0$ to $x=\frac{\pi}{2}$. Provide a graphical illustration.
\item The sum of the areas of rectangles under the curve of width 0.01
from $x=0$ to $x=\frac{\pi}{2}$. Provide a graphical illustration.
\item The area under the curve from $x=0$ to $x=\frac{\pi}{2}$.
\item The sum of the areas of rectangles under the curve of width 0.2
  from $\frac{\pi}{6}$ to $\frac{2\pi}{3}$. Provide a graphical illustration.
\item The sum of the areas of rectangles under the curve of width 0.01
from $\frac{\pi}{6}$ to $\frac{2\pi}{3}$. Provide a graphical illustration.
\item The area under the curve from $\frac{\pi}{6}$ to $\frac{2\pi}{3}$. 
\end{enumerate}

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