A really large staircase

Calculus Level 4

Ryan didn't believe me when I said that the harmonic series was divergent, so I took him to the infinitely long and infinitely tall staircase conveniently located a couple blocks from my house. This was a very special staircase. The \(n\)th step of the staircase had a height and width of \(\frac{1}{n}\) foot. Because BrillCo, the creators of the staircase, did not want to waste resources by making every step connect to the Earth, they made the staircase so that the bottom of the stairs formed a straight line at an angle of \(45˚\) to the horizontal. While Ryan trudged his way up the staircase, I decided to perform the non-infinite task of finding the volume of the staircase. If the staircase is straight and I measured the width of the staircase to be \(3\) feet, then the total volume of the staircase that I correctly calculated can be represented by \(N\). What is \(\lfloor100N\rfloor\)?

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