
\centerline{\bf Project: Change of Variables} \medskip

\centerline{Due Nov. 16}

\bigskip

Before reading further you should read section 15.8, Focus on
Theory, in our book.

Now that you have read 15.8, I will make a few comments on two
typical situations where one makes a change of variable.

\noindent {\bf Situation 1.} When the region is inconvenient in
the original variables, sometimes a change of variable will lead
to a much more convenient region. You have already seen how this
works with polar coordinates. Here is another simple example: let
{\bf R} be the region below. \vskip 1.2in

\noindent While this region can be described by two integrals in
the $x, y$ plane, we can transform it to an even easier region in
the $u,v$ plane by the change of variables $x=(u+v)/2, y=(v-u)/2$.
Try integrating
$$\int_{R} xy\ dA$$
with this change of variable, but you don't need to hand it in.

\noindent {\bf Situation 2.} Sometimes the integrand is really
messy in the original variables, and a change of variables will
simplify it quite a bit. For instance, suppose we wish to
integrate
$$\int_{R} \sin(xy)\cos(x+y)dA.$$
Without a change of variable, this is very difficult. An obvious
change of variable to try is $u=xy,v=x+y$. Of course, this change
of variable might transform the region of integration into
something uglier than the original region, but at least there
would be a fighting chance of evaluating the integral.

Now compute the following integrals using a change of variable.
You can of course check these integrals numerically if you'd like:
\item{1.} Evaluate
$$\int_{\bf R} x^2y^2dA,$$
where $\bf R$ is the region in the first quadrant between the
curves $y=1/x,y=2/x,y=x,y=3x$.
\item{2.} Let $\bf R$ be the ellipsoid
$$14x^2+4y^2+10z^2-8xy+16xz=36.$$
Use a change of variable to evaluate
$$\int_{\bf R}(x-z)dV.$$
Hint: See what happens with the substitution
$$u=x-y,v=x-z,w=x+z.$$
\item{3.} Evaluate
$$\int_{\bf R} e^{{y-x}\over{y+x}}dA,$$
where ${\bf R}$ is the region inside the triangle with corners
$(0,0),(1,1)$ and $(2,0)$.

\bye

Evaluate
$$\int_{\bf R} x^2y^2dA,$$
