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How does one find the area of this?

Hello everyone! The relation of the image above is \(x^2+y^2 = 2^y\). How do you find the area of the egg shape?

*Note: There is a section above the droplet, which should not be accounted for.

Made with Desmos graphing calculator

Note by Timothy Cao
1 month ago

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A different question about this graph: Consider graphs of the form \(x^2+y^2=a^y\). If \(a=2\), the graph is as above. If \(a=2.09\), the egg shape is now connected to the upper segment of the graph but if \(a=2.08\), the egg shape is still distinct. Therefore, at what value of \(a\) does this egg first connect to the upper part of the graph?

Kaan Berk Erdogmus - 2 weeks, 4 days ago

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Ayush Jain - 2 days, 20 hours ago

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Ayush Jain - 2 days, 20 hours ago

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@Ayush Jain SVC

Ayush Jain - 2 days, 20 hours ago

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@Ayush Jain jnlkn

Ayush Jain - 2 days, 20 hours ago

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Ayush Jain - 2 days, 20 hours ago

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Actually, that question has a precise answer: \(a = e^{2/e} \approx 2.0871\).

\[x^2 + y^2 = a^y.\]

At the transition point, the tip of the "egg" and the dip in the rest of the graph meet each other; due to symmetry, this must happen somewhere on the line \(x = 0\), and the connection will have a vertical tangent. Thus we set \(dx = 0\) in the total derivative and substitute \(a^y = y^2\):

\[2x\:dx + 2y\:dy = (\ln a) a^y\:dy\] \[2y = (\ln a) a^y = (\ln a)y^2.\] \[\ln a = \frac 2 y\ \ \ \therefore\ \ \ a = e^{2/y}.\]

Now substitute \(x = 0\) and this result in the original equation: \[y^2 = a^y = (e^{2/y})^y = e^2;\] \[y = e,\ a = e^{2/e}.\]

Arjen Vreugdenhil - 2 weeks, 4 days ago

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You can also generalize this solution for graphs of the form \(x^{2n}+y^{2n} = a^y\). The general solution for the critical \(a\) is \(a = e^\frac{2n}{e}\)

Kaan Berk Erdogmus - 2 weeks, 3 days ago

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It is approximately 2.316, but I think you will have to resort to numerical integration as I did. Formally, we have \[A = \int_{-c}^2 2\sqrt{2^y - y^2}dy,\] where \(c \approx 0.767 > 0\) is such that \((\tfrac12)^c = c^2\); however, there is no hope to solve this integral algebraically.

Arjen Vreugdenhil - 3 weeks, 6 days ago

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