Question: How big must r be chosen in the diagram, in order for the red area to equal one half of the area of the circle?
The above
problem was first posed in 1748, and it has been known since then that r is approximately equal to 1.1578. Apparently it just took us 272 years to figure out
an exact solution to it. Namely,
where the integrals are
contour integrals.
Good job, humanity.
In all seriousness, it's fascinating how difficult triviallooking math problems can become once you start to experiment a little with the functions/constants/boundary conditions/shapes/etc. Another good example of this is the calculation of the arc length of a curve.
Specifically, if y = f(x) is a continuously differentiable function on the interval (a,b), then its arc length s between a and b is
There are some simple exact solutions to the above integral, but let's just say that there are reasons why most calculus textbooks stick to examples where f(x) describes either a line, parabola, circle or hyperbolic cosine curve...
https://en.wikipedia.org/wiki/Arc_length
Quote:
In most cases, including even simple curves, there are no closedform solutions for arc length and numerical integration is necessary.

"Stephen Wolfram is the creator of Mathematica and is widely regarded as the most important innovator in scientific and technical computing today."  Stephen Wolfram