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Find the area enclosed by the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (Using green's Theorem)

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I am confused why we didn't use double integral here although answer makes sense without using double integral but I really don't understand why .

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Well, yeah, you could brute force it by figuring out the right bounds and doing a double integral, but then you wouldn't be attacking the problem with Green's theorem. –  Muphrid Dec 16 '12 at 3:57
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Green's theorem can be used to express the area of a region in terms of a line integral (the one on the left of your displayed equation). See the bottom of the page here for the derivation. The calculation in your post is arguably easier than using a double integral to find the area. –  David Mitra Dec 16 '12 at 3:58
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It may have helped you see the connection to the (typical) formal statement of Green's Theorem---as relating a double integral over $D$ to a line integral around the bounding curve of $D$---if they would have written the double integral first and then converted that to the line integral they began with. But alas...

See (3) here or the explanation here for why your text started with the integral they did.

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