Project: Change of Variables

Due Nov. 16

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. 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 R be the region below.

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

∫_{R} xy dA

with this change of variable, but you don’t need to hand it in. 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

∫_{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:

1. Evaluate

∫_{R} x^{2}y^{2}dA,

where R is the region in the first quadrant between the curves y = 1/x, y = 2/x, y = x, y = 3x.

2. Let R be the ellipsoid

14x^{2} + 4y^{2} + 10z^{2} − 8xy + 16xz = 36.

Use a change of variable to evaluate

∫_{R}(x − z)dV.

Hint: See what happens with the substitution

u = x − y, v = x − z, w = x + z.

3. Evaluate

∫_{R} e^{\frac{y−x}{y+x}} dA,

where R is the region inside the triangle with corners (0, 0), (1, 1) and (2, 0).