10217

# Double Integrals by Summing Values of a Cumulative Distribution Function

Let be a function, and suppose that its "cumulative distribution function" , is known. is the integral of over the rectangle below and to the left of , and the double integral of over a rectangle can be computed easily in terms of the values of at the corners via:
.
Checking boxes causes a region to be shaded such that the combination values at the checked corners is the integral of over the shaded region.

### DETAILS

Snapshot 1: only the vertex is chosen in the linear combination
Snapshot 2: both vertices and are chosen in the linear combination; light green regions are where the function's double integral was added only once in the linear combination with a positive coefficient; dark green regions are where the function's double integral was added twice in the linear combination
Snapshot 3: Both vertices C and D are chosen in the linear combination; light red regions are where the function's double integral was subtracted only once in the linear combination; dark green regions are where the function's double integral was subtracted twice in the linear combination
More information regarding this algorithm is available in Wikipedia, and in the following papers:
[1] F. C. Crow, "Summed-Area Tables for Texture Mapping," in SIGGRAPH '84: Proceedings of the 11th Annual Conference on Computer Graphics and Interactive Techniques, 1984 pp. 207–212.
[2] P. Viola and M. Jones, "Robust Real-Time Object Detection," International Journal of Computer Vision, 57(2), 2002 pp. 137–154. http:\\research.microsoft.com/en-us/um/people/viola/pubs/detect/violajones_ijcv.pdf

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.