Activity 12.9.4

Consider the problem of finding the area of the region $D'$ defined by the ellipse $x^2 + \frac{y^2}{4} = 1$. In this activity, we will look at what happens when we use a change of variables so that the pre-image of the domain is a circle.

1. Let $x(s,t) = s$ and $y(s,t) = 2t$. Explain why the pre-image of the original ellipse (which lies in the $xy$ plane) is the circle $s^2 + t^2 = 1$ in the $st$-plane.

2. Recall that the area of the ellipse $D'$ is determined by the double integral $\iint_{D'} 1 \, dA$. Explain why $$\begin{gathered} \iint_{D'} 1 \, dA = \iint_{D} 2 \, ds \, dt \end{gathered}$$ where $D$ is the disk bounded by the circle $s^2 + t^2 = 1$. In particular, explain the source of the “2” in the $st$ integral.

3. Without evaluating any of the integrals present, explain why the area of the original elliptical region $D'$ is $2\pi$.