%
%Homework on the Fundamental Theorem
%

\begin{homework}\
\label{homework:18}

\medskip

\exercise %1.
Find $d/du$ of the following functions of $u$:

\begin{centering}
\begin{tabular}{lll}
(a) $\disp\int_0^u\sin^2x\,dx$; &
(b) $\disp\int_{-1}^u\sqrt{1+t+t^2}\,dt$;&
(c) $\disp\int_u^\pi\sqrt{1+\sin^2y}\,dy$; \\
(d) $\disp\int_{-u}^u\sin^2x\,dx$;&
(e) $\disp\int_{-u}^u\sin^3x\,dx$.&
\end{tabular}\\
\end{centering}

\smallskip

\exercise %2.
Find $d/dt$ of the following functions of $t$:

\begin{centering}
\begin{tabular}{ll}
(a) $\disp\int_1^{a+bt}\frac{dx}{ x}$;& 
(b) $\disp\int_0^{-\frac{1}{ 2}gt^2+v_0t+s_0} \sin^2y\,dy$; \\
(c) $\disp\int_{t^2}^{t^3}\cos(x^2)dx$;&
(d) $\disp\int_{a+bt}^{A\sin(\alpha(t-C))}\sqrt{1+u^2}du$.
\end{tabular}
\end{centering}

\smallskip

\begin{rightindent}{3.5in}
\exercise %3.
The stage of a puppet show has its upper edge defined by $y=20+10\cos^2(0.1
\pi x)$ (the units are cm).  Its left side is the $y$-axis and its lower
edge is the $x$-axis (see the drawing).  After the show the curtain
closes from left  to right at 2 cm/sec.  (a) Write a definite integral for
the curtain's area as perceived by the audience after $t$ seconds. (b) What
is the rate of change of the curtain's area at time $t$?
\end{rightindent}
\hfill
%--- figure SB (Fundamental Theorem  Problem 3)--------------------
\begin{psfigure}{1.95in}{1.4in}{figSB.ps}
\end{psfigure}
%------------------------------

\smallskip


\exercise %4.
The same as Problem 3, except that now the upper edge is defined by
$y=40/\left(1+(0.01x-1)^2\right)$.

\smallskip

\noindent
\begin{rightindent}{3.0in}
\exercise %5.
The opening to a stage has its highest point 48 feet above the stage, it is
symmetrical about the center line (which we'll take as the $y$-axis), and
it is bounded by the flat stage and the curve $x=\pm 48\cos(\pi y/96)$ (see
the drawing).  A curtain with flat bottom is lowered by attaching two
heavy weights, one at each end, and letting them fall (with acceleration 32
ft/sec${}^2$), bringing down the curtain.  
\end{rightindent}
\hfill
%--- figure SC (Fundamental Theorem  Problem 5)--------------------
\begin{psfigure}{2.45in}{1.5in}{figSC.ps}
\end{psfigure}
%------------------------------
(a) Write a definite integral
for the amount of the stage blocked by the curtain after $t$ seconds.  (b)
At time $t$, what is the rate of change of the area blocked by the curtain?

\smallskip
\noindent

\exercise %6.
A flat vertical plate is moving from left to right (back and forth) across
a water pipe 1 cm in radius.

\begin{rightindent}{3in}
(a) Write a definite integral for the
area blocked off by the plate when it is $u$ cm to the right of center (the
area shaded in the drawing). 
(b) Suppose that $u$ is a sinusoidal function
of time.  If $u$ starts at $-1$ when $t=0$ and sweeps back and forth once
every 2 seconds, find a formula for $u(t)$.  (c) Find a formula for the
rate of increase of the area blocked (in cm${}^2$/sec) at time $t$.
\end{rightindent}
\hfill
%--- figure SD (Fundamental Theorem  Problem 6)--------------------
\begin{psfigure}{2.45in}{1.5in}{figSD.ps}
\end{psfigure}
%------------------------------

\end{homework}

