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Contents Index Calc Dark Mode Prev Up Next \(\newcommand{\R}{\mathbb{R}}
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Activity 12.8.4 .
Consider the problem of finding the area of the region
\(D'\) defined by the ellipse
\(x^2 + \frac{y^2}{4} = 1\text{.}\) 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.
(a) Let
\(x(s,t) = s\) and
\(y(s,t) = 2t\text{.}\) 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.
(b)
Recall that the area of the ellipse \(D'\) is determined by the double integral \(\iint_{D'} 1 \, dA\text{.}\) Explain why
\begin{equation*}
\iint_{D'} 1 \, dA = \iint_{D} 2 \, ds \, dt
\end{equation*}
where \(D\) is the disk bounded by the circle \(s^2 + t^2 = 1\text{.}\) In particular, explain the source of the “2” in the \(st\) integral.
(c) Without evaluating any of the integrals present, explain why the area of the original elliptical region
\(D'\) is
\(2\pi\text{.}\)
