It’s Newtonmas 2017 tomorrow!
What better way to celebrate than talk about integration!
The Batman Shape (sometimes called the Batman Curve, somewhat erroneously, I think) looks like this:
You can find details about it at Wolfram MathWorld, including its area in closed form. What’s interesting is integrating it without algebra, numerically. Quadrature would be difficult and a lot of work. Monte Carlo integration, not so much. What’s interesting is the relation between that and use of Monte Carlo integration for Bayesian computation, such as in Markov Chain Monte Carlo and slice sampling.
I was going to do the area of the Batman Shape using Monte Carlo integration and compare it with the exact value, but Jame Schloss has already done that. I might come back and do it using slice sampling, which will be a little interesting since it’s a two-dimensional figure. I’ll append that below if I do.