\begin{homework}\
\label{homework:5}

\medskip
\exercise
Suppose that at time $t=10$ sec an object is traveling at 30.0 m/sec.
Its acceleration is measured at two-second intervals until time $t=20$,
with the following results (the units of acceleration are m/sec${}^2$):


\begin{center}
\begin{tabular}{rlllllllll}
$t$      &10&12&14&16&18&20\\
$a(t)$  &2.32&2.43&2.52&2.59&2.65&2.70
\end{tabular}\\
\end{center}

\noindent
Use the trapezoid rule to find the velocity of the object at time $t=20$.

\smallskip
\exercise
Using $n=4$ subdivisions, see how close to the true value of
$\int_0^{\pi/2}\sin x\,dx$ you get using 
(a) the left endpoint Riemann sum,
(b) the right endpoint Riemann sum, 
(c) the trapezoid rule,
(d) Find how much closer you get to the true value using the trapezoid
rule with $10$ rather than 4 subdivisions.

\smallskip
\exercise
Find the arclength of the curve $y=x^3$ from $x=0$ to $x=2$ using
the trapezoid rule with $n=4$ subdivisions.

\smallskip
\exercise
Find the arclength of the ellipse $4x^2+y^2=4$ between the points
$(0,2)$ and $(0.5,\sqrt{3})$ by 
(a) solving for $y$ as a function of $x$
and applying the trapezoid rule with $n=5$ subdivisions to
$\int_a^b\sqrt{1+(f'(x))^2}dx$; and 
(b) writing the ellipse parametrically,
finding the values of $\theta$ corresponding to the two points $(0,2)$ and
$(0.5,\sqrt{3})$, and applying the trapezoid rule with $n=5$ subdivisions to
the integral for the arclength of a parametrized curve.

\smallskip
\exercise
In Example 3 of the last section, suppose you wanted to find the
distance traveled by the point $P$ during the 5 seconds from when the wheel
leaves the roof to when it hits the ground.  
(a) Explain why, even with the trapezoid
rule, you cannot expect to get an accurate answer with a small
value of $n$ such as $n=10$.  Now suppose that we are interested only in
the first 0.04 sec.  Use the trapezoid rule with $n=4$ subdivisions to find the
distance traveled during the first 0.04 sec by (b) the center of the wheel,
and (c) the point $P$.

\smallskip
\exercise
The table below shows the horizontal and vertical velocities of a
particle at 1-second intervals for the first 8 seconds.  Use the trapezoid
rule to find 
(a) the straight-line distance between the particle's location at time
$t=0$ and at time $t=8$, and
(b) the actual distance traveled along its path between $t=0$ and $t=8$.

\begin{center}
\begin{tabular}{rlllllllll}
$t$      &0&1&2&3&4&5&6&7&8\\
$dx/dt$  &7.02&6.59&5.96&5.23&4.40&3.67&2.74&1.81&1.08\\
$dy/dt$  &4.12&5.21&5.84&6.15&6.24&6.23&6.09&5.80&5.45
\end{tabular}\\
\end{center}

\end{homework} 
