Change of Variables — Question 3

Question 3 – Solution

Strategy. First remove the unequal axis scales, then exploit radial symmetry on the resulting unit disk.

Step 1: Linear scaling

The map $x=3u$, $y=2v$ sends the unit disk $D$ to $E$, with $$\left|\frac{\partial(x,y)}{\partial(u,v)}\right|=6, \qquad \frac{x^2}{9}+\frac{y^2}{4}=u^2+v^2.$$

Step 2: Polar coordinates in $D$ Write $u=r\cos\theta$, $v=r\sin\theta$ and $du\,dv=r\,dr\,d\theta$. Then $$\begin{align*} I&=6\int_0^{2\pi}\int_0^1r^2(r\,dr\,d\theta)\\ &=6(2\pi)\left[\frac{r^4}{4}\right]_0^1 =\boxed{3\pi}. \end{align*}$$

Verification The ellipse area is $6\pi$, and the transformed radial-square function has average $1/2$ on the unit disk. Thus the integral must be $(6\pi)(1/2)=3\pi$.