Partitions and Riemann Sums

Real Analysis · Axiom Academy

LESSON Partitions and Riemann Sums From slicing an interval to squeezing the exact area: how partitions, sample points, and refinement build the Riemann integral. 1. A Partition Slices the Interval A partition P of [a,b] is a finite set of cut points . It chops [a,b] into n subintervals [x_ i-1 , x_i] , the i -th one having width . The animation cuts [0,2] at five points and brackets each piece. 2. A Riemann Sum Adds the Rectangles On each slice, pick any sample point and raise a rectangle of height f(x_i^*) and width . Summing their signed areas gives a Riemann sum . The animation drops a rectangle on each subinterval of and tallies the running total. 3. Left, Right, and Midpoint Sums Three standard sample choices, shown side by side on the same partition of [0,2] with n=4 . Watch where each rectangle touches the curve: the left sum hugs the low edge of every rising slice, the right sum the high edge, and the midpoint splits the difference. 4. Upper and Lower Sums Trap the Area Instead of choosing a sample point, take the extremes. On each slice the upper sum uses the curve's highest value (the supremum) and the lower sum its lowest (the infimum). Every Riemann sum on that partition lands between them — the true area is trapped in the gap . The animation draws the upper rectangles down to the lower ones and shades the slack. lower sum every Riemann sum upper sum 5. Refine the Mesh, Recover the Integral

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