Laplace Convolution proof - can someone please explain the diagram

[thomas49th]

Homework Statement

http://gyazo.com/22f5c655071e49adf3f4652e8042873b


I derived the proof myself, but by arbitrary changing the t limit to infinity. I'm hoping that by understanding the diagram in the link above I will have a better idea of why I changed the limit. I currently look at it as we are finding the area of region R, but what does that have to do with signals f and g?

Thanks
Thomas

 

[thomas49th]

also found a reference here

http://math.fullerton.edu/mathews/c2003/laplacetransform/LaplaceConvolutionTheorem.1.pdf

Still not really getting the point of the diagram