Riemann Sum Calculator – Left, Right & Midpoint

A Riemann sum chops [a, b] into n rectangles of width Δx and adds their heights. For ∫₀¹ x² dx with n = 4: Δx = 0.25, and the right endpoints 0.25, 0.5, 0.75, 1 give heights 0.0625, 0.25, 0.5625, 1 — sum 1.875, times 0.25 = 0.46875, overshooting the exact 1/3 because x² rises. Left endpoints (0, 0.25, 0.5, 0.75) give 0.21875, an equal-footing underestimate; midpoint sampling (0.125, 0.375, 0.625, 0.875) gives 0.328125, already within 1.6% of 1/3. The calculator tabulates every rectangle, compares the three sampling rules, and shows the convergence as n grows — at n = 100 the midpoint rule lands within 0.00002. The left/right bracket [underestimate, overestimate] for monotone functions doubles as a rigorous error bound, and the sum itself is the definition of the definite integral in the limit n → ∞.

1. 1

   Enter f(x), a, b, and n

   The integrand, bounds, and rectangle count

2. 2

   Pick the sampling rule

   Left, right, or midpoint heights

3. 3

   Build the table

   Each rectangle’s xₖ*, height, and area is listed

4. 4

   Compare and converge

   All three rules plus the exact value when known

• 1

  Rising functions: left sum underestimates, right overestimates

• 2

  Midpoint errors are about a quarter of trapezoid errors

• 3

  The integral is defined as the n → ∞ limit of these sums

• Using (b−a)/n with n+1 points and off-by-one widths

• Sampling at interval edges for the midpoint rule

• Forgetting the final multiplication by Δx

• Averaging left and right and calling it midpoint — that is the trapezoid rule