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.

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## Comments

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TopNewestA 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?

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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: $y^2 = a^y = (e^{2/y})^y = e^2;$ $y = e,\ a = e^{2/e}.$

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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 $a = e^\frac{2n}{e}$

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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.

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