Lecture 8 - Distributed Ray Tracing
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replacement for slides covering integration.
follow reasoning in Appendix A of photon mapping book.

1) Integration.
   Show area under curve.
   Show quadrature.

2) Alternative -- randomly sample.
   Show this.
   Weight evenly.

   Converges slowly (actually error goes as 1/n^2) if function
   we're integrating has uneven distribution, esp. lots of zero areas.

3) Can we do better?
   In particular, if we know something about the function, can we do better?

4) Answer: yes.
   Importance sampling - use a-priori probability distribution.
   Weight each sample by inverse of probability.
   
   [illustrate this with quadrature - sample width is proportional
    to sampling density -- weight accordingly]

5) What you get:
   Expected value with less error; less expected variance.
   (i.e. pixel-to-pixel noise).