Скачать или смотреть Definite Integral fractional part of x with non integer limits, x-i = {x} in interval [i, i+1)

This fractional part("funny set notation") definite integral was motivated by the 0 to 4 limits of integration video produced by BlackPenRedPen(Nov 24, 2018) and by the highly instructive comment of math talent Angel Mendez Rivera(@MrBeen992 a huge admirer) ‪@angelmendez-rivera351‬ who formulated a ceiling floor methodology to evaluate the integral. She prefers the x-floor(x) representation of the function {x}. KohuGaly also shared some insightful commentary from a computer science perspective! Rivera is a highly principled human being who is direct to the point of being curt/abrupt/ brusque/abrasive, but a fair person.

Rivera generalizes the problem to handle arbitrary non-integer upper and lower limits of integration. The area underneath the curve often involves trapezoids and smaller triangles. Both methods involve breaking up the integral based on the discontinuity of fractional part function at each integer.

The method I used involves observing that x-i = {x} in interval [i, i+1) for all integers. The antiderivative of x-i is simple, so this method can also evaluate any definite integral with any real number upper and lower limits of integration. The area of integration, in this particular case, is a trapezoid and a triangle.