question on Lebesgue integration

[wazen]

how can i solve

 

 

by Lebesgue integration

 

??????

[matt grime]

You seem to have the wrong impression of Lebesgue integration. It doesn't actually tell you how to integrate something like that, it just defines an integral for more functions than Riemann integration.

 

That function is continuous, so it is Riemann integrable. Finding what the integral is is a different matter, but is a very common one. I'm sure you can find it on the internet: it is the Bell curve, or Gauss curve, or the Normal distribution's pdf after all.

 

The simplest way is to define that integral as I(x), and consider I(x)I(y), then do a change of variable to polar coordinates.

 

Nothing to do with measure theory at all.