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\title{ESP Workshop, Worksheet \#22}
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{\bf \Large ESP Workshop, Worksheet \#22}

{\bf \large Thursday \today}

{\bf \large AI: Eric Katerman}
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\vspace{.15in} 

One of the most important skills to learn from this course---and one
that will certainly be tested on either the next exam or the
final---is to be able to correctly {\bf identify limits of integration
  for double and triple integrals}. Professor Durbin described this a
bit yesterday for triple integrals, and he admitted to me that he
didn't present it the way he should have.  Let's learn the easy way to
identify regions of integration.  For each of the following integrals,
follow these steps:
\begin{enumerate}
\item If a region $D$ is not given to you explicitly, write down the
  region in set notation: eg. $D = \{(x,y) : -1 \leq x \leq 1, -x
  \leq y \leq x^2 \}$
  
\item Draw the region in the $xy$-plane (for double integrals) or in
  $xyz$-space (for triple integrals).
  
\item Work from the ``inside-out'' to check your picture: eg. if you
  were integrating $$\iiint _D f(x,y,z) dx\ dy\ dz$$, draw the
  projection of your picture from part (b) onto the $yz$-plane, and
  then onto the $z$-axis.
  
\item Write down all possible rearrangements of the integral, with
  the proper bounds; for double integrals, there are two
  possibilities ($dx\ dy$ and $dy\ dx$), and for triple integrals,
  there are six (which ones?). To do this, take your picture from
  part (b), and follow the same steps as in part (c). For example,
  if now we were doing $\iiint _D f(x,y,z) dz\ dx\ dy$, we would
  first project onto the $xy$-plane and then onto the $y$-axis.
  
\item (Optional.) Evaluate a few of the rearrangements to check your
  answers.
  
\end{enumerate}

See? It's a simple four-step process. (NB: you won't always be able to
do each of these steps---the obstruction has to do with type I vs.
type II regions and so forth. For more on that, see section 16.7 in
the text.) Now let's practice!
  
\begin{enumerate}
\item $\int _0 ^1 \int _{3y} ^3 e^{x^2}\ dx\ dy$
\item $\int _0 ^9 \int _0 ^{\sqrt{x}} y \cos (x^2)\ dy\ dx$    
\item $D = \{(x,y,z): x^2 + z^2 = 4,\ y=0,\ y=6 \}$
\item $\int _0 ^1 \int _y ^1 \int _0 ^y (xyz)\ dz\ dx\ dy$
\item $\int _0 ^1 \int _0 ^{1-x^2} \int _0 ^{1-x} (xy +yz)\ dy\ dz\ dx$

\item (Challenge!) In this problem, we will find the volume enclosed
  by a hypersphere in $n$-dimensional space!  Oooooh!
\begin{enumerate}
\item Use a double integral and trig substitution, together with the formula:
\begin{displaymath}
\int \cos ^2 u\ du = \frac12 u +\frac14 \sin 2u +C
\end{displaymath}
to find the area enclosed by a circle of radius $r$.

\item Use a triple integral and a trig substitution to find the volume
  inside a sphere with radius $r$.

\item Use a quadruple integral to find the hypervolume enclosed by the
  hypersphere $x^2 +y^2 +z^2 +w^2 = r^2$ in $\mathbb{R}^4$. Hint: use
  trig substitution and the following reduction formulas:
\begin{eqnarray*}
\int \sin ^n x\ dx &=& -\frac1n \sin ^{n-1} x\ \cos x +\frac{n-1}{n}\int \sin ^{n-2} x\ dx \\
\int \cos ^n x\ dx &=& \frac1n \cos ^{n-1} x\ \sin x +\frac{n-1}{n}\int \cos ^{n-2} x\ dx
\end{eqnarray*}

\item Use an $n$-tuple integral to find the volume enclosed by a
  hypersurface of radius $r$ in $n$-dimensional space $\mathbb{R}^n$.
  Hint: the formulas are different for $n$ even and $n$ odd.

\end{enumerate}

\end{enumerate}  



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