\begin{homework}\
\label{homework:6}

\medskip

\exercise
Find the coordinates of the center of mass of a uniform flat plate that
is bounded by: (a) the line $x+2y=6$, the $x$-axis, and the $y$-axis; (b)
the parabola $y=x^2$, the $x$-axis, and the line $x=2$; (c) the right half
of the circle $x^2+y^2=r^2$ and the $y$-axis; (d) the top half of the
ellipse $x^2+4y^2=4$ and the two lines joining the point $(0,-4)$ to the
points where the ellipse meets the $x$-axis.\footnote{To compute the area
of the plate in part (d), use the following general principle:
If a shape such as the unit circle is expanded horizontally by a factor of
$a$ and vertically by a factor of $b$, then its area is multiplied by a
factor of $ab$.}

\smallskip
\exercise
Find the centroid of Nevada.  Suppose that Nevada is a trapezoid with
corners at the four points

$42^\circ$ northern latitude \& $120^\circ$ western longitude,

$42^\circ$ northern latitude \& $114^\circ$ western longitude,

$39^\circ$ northern latitude \& $120^\circ$ western longitude,

$35^\circ$ northern latitude \& $114^\circ$ western longitude.

\smallskip
\exercise
Use the trapezoid rule with $n=6$ subdivisions to find the center of mass of
a uniform flat plate in the shape bounded by the curve $y=\tan x$, the
curve $y=-\tan x$, and the line $x=\pi/4$.

\end{homework} 

