So I was reviewing my random process notes. In it there is an integral that they have that I can't seem to get the right derivation of when they try to simply the math for ergodic mean. Basically, you have the following:(adsbygoogle = window.adsbygoogle || []).push({});

A square from (-T,T) on both the x-axis and y-axis. What they want to do is integrate over the diagonal lines u = x - y where u is a constant.

So rather than a double integral over dx and dy from (-T, T) for each, they now have one integral du from (-2T, 2T).

The part that I have a problem with is that the derivation of the area of the diagonal strip u = x-y which the textbook says can be shown to be (2T - u)du for u from 0 to 2T and (2T+u) for u from -2T to 0.

So the way I did it was to think of that diagonal strip as subtraction of two triangles. Which when I do the math ends up giving me the area as (2T-u)du - 1/2*(du)*(du). I tried to imagine this as the equation for the area of the strip if it were to be turned into a parallelogram, which means that the extra bit of area would be 1/2*(du)*du) which is why its subtracted and as du→0, it reduces to (2T-u)du. But the part where the base of the parallelogram equals (2T-u) just doesn't make sense.

I am wondering, what is the right way to do this derivation?

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# Area of a Square

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