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\begin{document}
\bc
\large\bf
MATH 119\\
Honors Calculus II\\
Chapter 5 
\ec

\bigskip
\bigskip
\lhwprob{1.}{
Inside the vertical strip $\,0 \leq x \leq \pi/2\,$ let $R$ be the region
bounded by the curves $\,y=\tan x\,$ and $\,y=\sin 2x$\@. \ Find the area
of $R$\@.
}

\bigskip
\bigskip
\lhwprob{2.}{
Find the volumes obtained by rotating the region of problem 1 about
each of the coordinate axes.
}

\bigskip
\bigskip
\lhwprob{3.}{
Find the area of the portion of the elliptical region
\[ \frac{x^2}{a^2} \ + \ \frac{y^2}{b^2} \ \leq \ 1 \]
for which
\[ x \ \geq \ \sqrt{a^2 - b^2} \]
when $a$ and $b$ are given positive constants with $\,a > b$\@.
}

\bigskip
\bigskip
\lhwprob{4.}{
Find the average value of the $x$-coordinate of a point in the region
bounded by the curves \ $x = 0$\@, \ $y = \sin x$\@, \ and
\ $y= \cos x$\@.
}

\bigskip
\bigskip
\lhwprob{5.}{
Find the volume of the solid whose cross-sections by planes
perpendicular to the $x$-axis, each of which is a square with side
of length $\, \sin x \,$, \ exist for $\,0 \leq x \leq \pi$\@.
}


\bigskip
\bigskip
\lhwprob{6.}{
Find the centroid of the first quadrant portion of the disk of
radius $a$ with center at the origin.
}

\bigskip
\bigskip
\lhwprob{7.}{Find the area of the surface obtained by rotating
the piece of the curve $\, y = \cosh x \,$ within the interval
$\, 0 \leq x \leq 2\,$ about the $x$-axis.
}


\bigskip
\bigskip
\lhwprob{8.}{Let $\, f(x) = \cosh x$\@. \ Find:
\begin{enumerate}
\item[(a)] \ the centroid of the curve segment that is the part of
             the graph of $f$ within the interval $\,0 \leq x \leq r$\@.
\item[(b)] \ the centroid of the region that lies between the graph
             of $f$ and the $x$-axis within the interval
             $\,0 \leq x \leq r$\@.
\end{enumerate}
}

\bigskip
\bigskip
\lhwprob{9.}{Find the centroid of the corkboard region: the region
in the first quadrant consisting of all points for which
\[ \sqrt{x} + \sqrt{y} \leq 1 \ . \]
}

\bigskip
\bigskip
\lhwprob{10.}{
Find the average value of the $y$-coordinate of a point in the region
bounded by the curves \ $x = 0$\@, \ $y = \sin x$\@, \ and
\ $y= \cos x$\@.
}

\bigskip
\bigskip
\lhwprob{11.}{
Find the area of the surface obtained by rotating the curve
\[ (x, \, y) \ = \ (\cos^4 t, \ \sin^4 t) \,, \ \ \ \ 0 \leq t \leq \pi/2 \]
about the line $\, x + y = 0$\@.
}

\bigskip
\bigskip
\lhwprob{12.}{
Find the centroid of the portion of the curve 
\[  x = \cos t \,, \ \ y = \frac{\cos 2t}{4}  \]
lying below the $x$-axis.
}


\end{document}
