\begin{homework}\
\label{homework:1}

\medskip

\exercise
Find the area between each of the following pairs of curves:

(a) $y=x^4-x^2$ and $y=x^2$ (the part on the right of the $y$-axis); 

(b) $x=y^3$ and $x=y^2$; 

(c) $y=x^n$ and $x=y^n$ (in the first quadrant), where $n>1$; 

(d) $x=1-y^2$ and $y=-x-1$;

(e) $x=3y-y^2$ and $x+y=3$; 

(f) $y=\cos(\pi x/2)$ and $y=1- x^2$ (in the first quadrant); 

(g) $y=\sin(\pi x/3)$ and $y=x$ (in the first quadrant).

\medskip 
\exercise
Find the ``triangular-looking'' area bounded by the three curves: 

(a) $y=\sin x$, $y=\cos x$, and $x=0$ (in the first quadrant); 

(b) $y=\sqrt{x}$, $y=20-x$, and $y=2$.

\medskip
\exercise
For each velocity function find both the net distance and the total
distance traveled during the indicated time interval (graph $v(t)$ to
determine when it's positive and when it's negative): 

\begin{centering}
\begin{tabular}{ll}
(a) $v=\cos(\pi t)$, $0\le t\le 2.5$;
&
(b) $v=-9.8t+49$, $0\le t\le 10$; 
\\
(c) $v=3(t-3)(t-1)$, $0\le t\le 5$; 
&
(d) $v=\sin(\pi t/3)-t$, $0\le t\le 1$.
\end{tabular}\\
\end{centering}

\medskip
\exercise
Two parallel charged plates create a force of constant
magnitude $F$ on a charged particle of mass $m$.  Suppose that the plates
are rotating at the constant 
angular velocity of $\omega$ radians/sec, so that the {\it direction} of
the force is {\it not} constant (see the figure at the right).  Assume that
the particle is at rest at the origin of the $xy$-plane at time $t=0$, and
that the force pulls in the positive $x$-direction at time $t=0$ 

\begin{rightindent}{3.5 truein}
\noindent
and rotates counterclockwise.  (Also assume that $\omega$ is small enough
so that the effects of the magnetic field created by the varying electric
field can be neglected.)  To describe the motion of the particle, consider
the $x$- and $y$-directions separately. 


(a) Express the $x$-component of the force vector in terms of the angle
$\theta$ through which the plates have rotated, and express the latter
angle in terms of $t$.  Then find the $x$-acceleration function and the
$y$-acceleration function (by dividing the force by the mass $m$).  Here
$F$, $m$, and $\omega$ are constants.

\end{rightindent}
\hfill
%-figure AE -----------------------------
\begin{psfigure}{2in}{1.5in}{figAE.ps}
%\putcrosshair{0,0}
\puttext{21., 5.}{\small$\theta$}
\end{psfigure}
%------------------------------


(b) Find the horizontal component of velocity and the vertical component
of velocity as functions of time.

(c) Find formulas for the $x$-coordinate of the particle at time $t$ and
for the $y$-coordinate of the particle at time $t$.

(d) Set the constants all equal to 1, and sketch the parameterized curve in
part (c).

(e) How is the trajectory affected if the particle has an initial velocity
$v_0$ in the $y$-direction?  For what value of $v_0$ (expressed in terms of
$F$, $m$, and $\omega$) will the particle travel in a circle?

\end{homework}

