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\title{Applied Calculus}
\author{Homework 8}
\date{Due in class, December 1, 2015}
\maketitle

\begin{enumerate}

\item Use Scientific Python for this question.

Measurements of the net photosynthetic activity $a_i$ of a patch of
forest are taken at times $t_i$ throughout one day.  A sample of such
a data set\footnote{The concrete values are made up, but the problem
is inspired by J.D. Lewis, D. Olszyk, and D.T. Tingey, \emph{Seasonal
patterns of photosynthetic light response in Douglas-fir seedlings
subjected to elevated atmospheric $\text{CO}_2$ and temperature}, Tree
Physiology \textbf{19} (1999), 243--252.} can be found as an Ipython
notebook at
\begin{center}
  \url{http://math.jacobs-university.de/oliver/teaching/jacobs/fall2015/esm106/notebooks/20151124-homework-data.ipynb}
\end{center}
where the $a_i$ are given in
$\mu\text{mol}\,\text{CO}_2\,\text{m}^{-2}\,s^{-1}$ and the times $t_i$
are given in hours.

Use the data to compute
\begin{enumerate}
\item The total $\text{CO}_2$ uptake in
$\mu\text{mol}\,\text{CO}_2\,\text{m}^{-2}$ over one day,
\item The total $\text{CO}_2$ uptake in
$\mu\text{mol}\,\text{CO}_2\,\text{m}^{-2}$ before 12 noon,
\item The total $\text{CO}_2$ uptake in
$\mu\text{mol}\,\text{CO}_2\,\text{m}^{-2}$ after 12 noon.
\end{enumerate}


\item Find the following indefinite integrals.
\begin{enumerate}
\item $\displaystyle \int x^2 \, \sqrt{x^3+1} \, \mathrm d x$
\item $\displaystyle \int \tan \theta \, \mathrm d \theta$
\item $\displaystyle \int t \, \sin t \, \mathrm d t$
\item $\displaystyle \int t^2 \, \sin t \, \mathrm d t$
\end{enumerate}
\hfill(2.5 points each)

\item Find the following definite integrals.
\begin{enumerate}
\item $\displaystyle \int_{0}^1 r \, \sqrt{1-r^2} \, \mathrm d r$
\item $\displaystyle \int_{-\pi}^\pi \sin \theta \, \mathrm d \theta$
\item $\displaystyle \int_1^2 y \, (y^2+1)^{3/2} \, \mathrm d y$
\item $\displaystyle \int_0^\infty x \, \mathrm{e}^{-x^2} \, \mathrm d x$
\end{enumerate}
\hfill(2.5 points each)


\item (From MLS, p.~470.)  For a particular town, new cases of flu are
reported at a rate
\[
  R(t) = 4 \, t \, \e^{-0.05 t} \,,
\]
where $R$ is measured in people per day and $t$ is time in days.  
\begin{enumerate}
\item If $500$ flu cases have been reported at $t=0$, what is the
estimated number of cases after 10 days?

\item How many cases will be reported over the entire epidemic?
\end{enumerate}

\item (From MLS, p.~470.)  The volume of a tree between heights $a$
and $b$ for a tree with total height $H$ may be approximated by
\[
  \int_a^b K \, (H-x)^{3/2} \, \mathrm d x \,,
\]
where $K$ is a constant.  Use this to approximate the total volume of
a tree, and note that your answer will include the constants $H$ and
$K$. 

\end{enumerate}




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