Definition of Riemann Integral

Real Analysis · Axiom Academy

LESSON Defining the Riemann Integral When do upper and lower approximations of an area agree? That agreement is the integral — and exactly when it exists, the function is integrable. 1. Trapping the Area From Both Sides Partition [a,b] into subintervals. On each piece [x_ i-1 , x_i] take the highest value the curve reaches, , and the lowest , . The tall rectangles sum to the upper sum U(P,f) ; the short ones to the lower sum L(P,f) . The true area is forever caught between them: . Upper sum — rectangles reach the peak of each piece Lower sum — rectangles sit at the floor of each piece 2. Integrability: The Gap Must Close Watch only the difference . The gap between the upper and lower sums is the strip the curve carves out of each rectangle — exactly . As the partition refines, that sliver shrinks. The function is Riemann integrable precisely when this gap can be made smaller than any tolerance . For a continuous f on the closed interval [a,b] , uniform continuity forces this gap to 0 as the mesh — so every continuous function is integrable, and its upper and lower sums both converge to the single number . For the curve shown, the true area is . Doubling the number of subintervals roughly halves the gap: U - L falls , squeezing the upper and lower sums onto that common value. 3. When They Never Meet: Not Every Function Integrates

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