Having the two functions $f(x) = x^2 + \sqrt{16-x^2}$ and $g(x) = x^2 - \sqrt{16-x^2}$
plotted as
how to find out the area of the enclosed area which looks like a "mouth" in an elegant way?
My thoughts
I've thought about adding $16$ to both functions, which should have no effect on the integrals of the two functions, but is this the most elegant way?
I've come up with 16 by observing that the lower curve only touches the $x$ axis, anything else (like 15 or 17) makes it cut the $x$ axis, which complicates things.
But why is $16$ the right number?
With $+16$, the two functions look like this:
If $+16$ is the most elegant technique, how to proceed afterwards?