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\begin{center}Math 132: Calculus for Physical and Life Sciences 2 \\ Problem Set 5 \\ Due Friday, February 29, 2008, at the beginning of class.\end{center}\vspace{.1in}

General Directions: You must show all work for credit on these problems. \vspace{.1in}

\begin{enumerate}
\item Explain why each of the following
integrals is an improper integral and determine whether
each one converges or diverges.  Evaluate those that are convergent.  \vspace{.1in}

\begin{tabular}{ll}

(a) $\displaystyle \int_4^{\infty} \frac{1}{x^{3/2}}\, dx$ & 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (b)  $\displaystyle \int_0^4 \frac{1}{\sqrt{4-x}}\, dx$\\ \ & \\
(c) $\displaystyle \int_0^2 \frac{x}{x^2-1}\, dx$&
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (d) $\displaystyle \int_e^{\infty} \frac{1}{x \ln x}\, dx$\\ \ & \\
(e) $\displaystyle \int_0^{\infty} x e^{-x^2}\, dx$&
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (f) $\displaystyle \int_0^{\infty} \sin^2 x\, dx$\\ \ & \\
(g) $\displaystyle \int_0^1 \frac{e^{\sqrt{x}}}{\sqrt{x}}\, dx$&
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \  (e)  $\displaystyle \int_{1/2}^1\frac{2}{\sqrt{1-x^2}}\, dx$

\end{tabular} \vspace{.1in}


\item (i) Sketch the given curves and determine where they intersect. \vspace{.1in}


(ii) Find the area bounded by the given curves. \vspace{.1in}

\begin{enumerate}

\item $y=-x^2$ and $y=x^2-2x$\vspace{.1in}


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



\item $ \displaystyle y=\frac{5}{x}+x$ and $y=6$ (in the region $x>0$) \vspace{.1in}

\item $y=\sqrt{x}$ and $y=x-2$, ($x>0$) \vspace{.1in}

\item $ y=e^x$, $y=e^{-x}$, $x=0$, and $x=1$\vspace{.1in}



\item $x=2-y^2$ and $x=y^2$
\end{enumerate}


\item Consider the curve $\displaystyle y=\frac{1}{x^2}$ for $1 \leq x\leq 6$. 

\begin{enumerate}
\item Calculate the area under this curve on the given interval. 
\vspace{.1in}

\item Determine $c$ so that the line $x=c$ bisects the area of part (a). (\emph{I.e.}, it divides the region into two regions of equal area.)\vspace{.1in}

\item Determine $d$ so that the line $y=d$ bisects the area of part (a). 

\end{enumerate}
\end{enumerate}

\vfill\eject
\noindent
{\it Selected Answers}
\vskip 10pt
Selected answers: 
\begin{enumerate}
\item[1.]
\begin{enumerate}
\item[(a)] Infinite interval of integration. Integral converges to $1$. 
\item[(c)] Infinite discontinuity (vertical asymptote) at $x=1$. Integral is divergent. 
\item[(e)] Infinite interval of integration. Integral converges to $1/2$. 
\item[(g)] Infinite discontinuity (vertical asymptote) at $x=0$. Integral converges to $2e-2$.  
\end{enumerate}
\item[2.]
\begin{enumerate}
\item[(a)] $1/3$
\item[(c)] $12-5 \ln 5$
\item[(e)] $e+1/e-2$
\end{enumerate}
\item[3.]  (c) $\displaystyle d=\frac{17}{12}-\frac{1}{3}\sqrt{15}$
\end{enumerate}

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