
\begin{homework}\
\label{homework:9}

\medskip
\exercise
Find the following improper integrals (some of them might be $\infty$):

\smallskip

\begin{tabular}{lll}
(a) $\disp\int_1^\infty dx/(x\sqrt{x})$; 
    &(e) $\disp\int_0^\infty dx/(1+x^2)$; 
        &(i) $\disp\int_{-10}^{10}dx/\sqrt{100-x^2}$; \\
(b) $\disp\int_0^1 dx/(x\sqrt{x})$;
    &(f) $\disp\int_{-\infty}^\infty dx/(1+x^2)$;
         &(j) $\disp\int_0^1 x\,dx/\sqrt{1-x^2}$, \\
(c) $\disp\int_1^\infty dx/\sqrt{x}$; 
    &(g) $\disp\int_{-\infty}^\infty dx/(9+x^2)$; 
        &(k) $\disp\int_0^\infty x\,dx/(1+x^2)$; \\
(d) $\disp\int_0^1 dx/\sqrt{x}$; 
    &(h) $\disp\int_0^1 dx/\sqrt{1-x^2}$; 
        &(l) $\disp\int_0^\infty x\,dx/(1+x^4)$.
\end{tabular}



\smallskip
\exercise
Let $n$ be a positive number.  Find (in terms of $n$) the volume of the
solid obtained by rotating around the $y$-axis the part of $y=-1/x^n$ below
$y=-1$ in the fourth quadrant.  For what values of $n$ is this volume
infinite?

\smallskip
\exercise
Suppose that $n>2$.  Find (in terms of $n$) the balancing point of the
uniform flat plate bounded by the graph of $y=1/x^n$, the graph of
$y=-1/x^n$, and the line $x=1$.  (The plate extends to the right all the
way out to infinity.)

\smallskip
\exercise
Show that the following uniform flat plates have finite mass but do not
have center of mass at a finite distance: 

(a) Problem 3 when $1<n\le 2$;

(b) the plate bounded by the graph of $y=1/(1+x^2)$, the graph of
$y=-1/(1+x^2)$, and the $y$-axis.  This means that you could lift up the
plate, but you could not balance it!

\end{homework} 

