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YZ
Numerade Educator

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Problem 12 Easy Difficulty

The area of an ellipse the area $\pi a b$ of the ellipse $x^{2} / a^{2}+y^{2} / b^{2}=1$ can be found by integrating the function $f(x, y)=1$ over the region bounded by the ellipse in the $x y$ -plane.
Evaluating the integral directly requires a trigonometric substitution. An easier way to evaluate the integral is to use the transformation $x=a u, y=b v$ and evaluate the transformed integral over the
disk $G : u^{2}+v^{2} \leq 1$ in the $u v$ -plane. Find the area this way.

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Video Transcript

Okay, So first we're gonna find a tree coping G if you envy which is equals to that X over W That's so weird that we now the way over W and dela y over dead will be fucking the result. A plug in the value we have zero a zero and zero p. So the answer is a p the area he goes to the exterior oi which is also be a over a region or will be transformed Overridden g. We have the UDV time surgery coping, which is a and B. So now the reason for you is from that one toe one for B recalled. This is a unit circle after transformation. So for we is that square with one Once you swear to positive square the promise you squared we have inside. Do you? Maybe I'm Stevie you So the answer will goes to a B times pi which is exactly the geometrical from there for our piglets was a and B. The airport is also a baby so that's two results agrees

YZ
University of Illinois at Urbana-Champaign
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