Calculating a Riemann Sum

Calculus 1 · Axiom Academy

EXAMPLE Calculating a Riemann Sum Work an integral from the definition: a right-endpoint sum, simplified with sigma formulas, then a limit. Compute directly from the definition — as the limit of right Riemann sums — and confirm the exact area under f(x)=x^2 between x=1 and x=3 . A right Riemann sum with n = 4 subintervals overestimates the area (each rectangle's height is taken at its right edge). Letting n → ∞ shrinks the gap to zero and gives the exact integral. Nicely done. You evaluated a definite integral straight from its definition — as a limit of Riemann sums. The sample point: for a right sum on [a,b] , and . The limit: as every 1/n term vanishes, collapsing R_n to the exact value . This limit of Riemann sums is exactly how the definite integral is defined — the rectangles become the area.

This is the written version of the interactive lesson above. See the full Calculus 1 course.