\begin{homework}\
\label{homework:2}

\medskip

\exercise
Graph both $v(t)$ and $s(t)$ for an object which (a) falls from rest
from a height of 44.1 m; (b) is thrown upward at 12.25 m/sec from a height
of 44.1 m; (c) is thrown downward at 12.25 m/sec from a height of 44.1 m.
In each case determine when it hits the ground.

\smallskip
\exercise
At what speed does an object dropped from rest from 100 meters hit the ground?
How long does it take to travel the last 10 meters?

\smallskip
\exercise
Suppose you look out a window of a skyscraper and see someone throw an
object downward.  Your window is at a height of 450 ft.  The object passes
your window after 2 seconds and hits the ground after 5 seconds (from the
time it was thrown).  Find the velocity at which it was thrown down, and
also the height from which it was thrown.  Take $g=32$ ft/sec${}^2$, and
neglect air resistance.

\smallskip
\exercise
Same as Problem 3, except that you did not see the moment when the
object was dropped.  However, in this problem you know that it fell from
rest (i.e., $v_0=0$), and this time you measure exactly 2 seconds between
the time the object passes your window (which is at a height of 450 ft) and
the time it hits the ground.  From what height did it fall?

\smallskip
\exercise
Find the volume of the solid obtained by revolving around the $x$-axis
the region bounded by: (a) $x+y=2$ and the $x$- and $y$-axes; (b) $y=x-x^2$
and the $x$-axis; (c) $y=\sqrt{\sin x}$, the $y$-axis, and the line $y=1$.

\smallskip
\exercise
Let $S$ be the region of the $xy$-plane bounded above by the curve
$x^3y=64$, below by the line $y=1$ and on the left by  the line $x=2$.  Find
the volume of the solid obtained by rotating $S$ around (a) the $x$-axis,
(b) the line $y=1$, (c) the $y$-axis, (d) the line $x=2$.

\smallskip
\exercise
Find (in terms of the constants $a$ and $b$) the volume of the solid
obtained by rotating the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ around
(a) the $x$-axis, (b) the $y$-axis.  (The resulting solids are called {\bf
ellipsoids}.)

\smallskip
\exercise
A hemispheric bowl of radius $r$ contains water to a depth $h$.  Find
the volume of water in the bowl.

\smallskip
\exercise
The base of a tetrahedron of height $H$ is an equilateral triangle of
side $s$.  Its cross-sections perpendicular to its altitude are
equilateral triangles.  Express its volume $V$ as an integral, and
find a formula for $V$ in terms of $H$ and $s$.  (Hint: Situate the
tetrahedron on its side with its vertex at the origin and its base
perpendicular to the $x$-axis at $x=H$. Use similar triangles to find
a side of the cross-section at $x$.)

\end{homework}
 
