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\begin{center}Math 132: Calculus for Physical and Life Sciences 2 \\ Problem Set 6 \\ Due Friday, March 14, 2008, at the beginning of class.\end{center}\vspace{.1in}

General Directions: You must show all work for credit on these problems. 

\begin{enumerate}
\item Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.  Sketch the region, the solid, and a typical disk or washer.
\begin{enumerate}

\item $y=x^2+2$, $x=1$, $x=2$, $y=0$; about the $x$-axis \vspace{.1in}

\item $y={1\over 2}x+3$, $y=x^2$; about the $x$-axis\vspace{.1in}

\item $y=\sin x$, $x=0$, $x=\pi$, $y=0$; about the $x$-axis \vspace{.1in}

\item $y=x^3$, $y=\sqrt{x}$, $x=0$, $x=1$; about the $x$-axis\vspace{.1in}

\item $y={1\over x}$, $y=x$, $y=2x$ for $x>0$; about the $x$-axis\vspace{.1in}

\item $x=4-y^2$, $x=y^2$; about the $y$-axis\vspace{.1in}

\item $x={1\over y}$, $y=4$, $x=2$; about the $y$-axis\vspace{.1in}

\end{enumerate}

\item The base of a certain solid is the circle $x^2+y^2=9$ and each cross section perpendicular to the $x$-axis is an equilateral triangle with one side across the base.  Find the volume of the solid.

\item A nose cone for a space reentry vehicle is designed so that a cross section, taken $x$ feet from the the tip and perpendicular to the axis of symmetry, is a circle of radius ${1\over 4}x^2$ ft. Find the volume of the nose cone given that its length is $20$ ft.

\item Graph the curve and find its exact length.

\begin{enumerate}
\item $x=4t+3$, $y=3t-2$,  $0\le t\le 2$ \vspace{.1in}

\item $x={1\over 3}t^3$, $y={1\over 2}t^2$, $0\le t\le 1$\vspace{.1in}

\item $x=3\sin (2t)$ and $y=3 \cos(2t)$, $0\le t \le {{\pi}\over 2}$ \vspace{.1in}

\item $y=3x^3+2$ and $2\le x \le 4$ \vspace{.1in}

\end{enumerate}

\item Use Simpson's Rule with $n=10$ to estimate the arc length of the curve $y=\ln x$ for $1\le x\le 4$.\vspace{.2in}

\item Find the average value of the function on the given interval. 

\begin{enumerate}
\item $y=x\cos(3x)$, $[0,{\pi}]$
\vspace{.1in}

\item $y=x\sqrt{1+3x^2}$, $[1,3]$\vspace{.1in}



\end{enumerate}

\item Find the number $b$ such that the average value of $f(x)=x^3-1$ on the interval $[0,b]$ is equal to $6$.

\item The amount of a certain drug present in a patient's body for the first $4$ days after the drug has been administered is $$C(t)=5e^{-0.2t}$$ units. Determine the average amount of the drug present in the patient's body for the first $4$ days after the drug has been administered.
\end{enumerate}



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