Riemann Sum Calculator Formula

Understand the math behind the riemann sum calculator. Each variable explained with a worked example.

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Formulas Used

Exact Val

exact_val = exp_val != -1 ? (1 / (exp_val + 1)) * (pow(b_val, exp_val + 1) - pow(a_val, exp_val + 1)) : 0

Rect Width

rect_width = dx

Error Est

error_est = abs(b_val - a_val) * abs(pow(b_val, exp_val) - pow(a_val, exp_val)) / (2 * num_rect)

Variables

VariableDescriptionDefault
exp_valExponent of x2
a_valLeft Bound0
b_valRight Bound1
num_rectNumber of Rectangles10
dxDerived value= (b_val - a_val) / num_rectcalculated

How It Works

Riemann Sums

Concept

A Riemann sum approximates the area under a curve by dividing the interval into rectangles:

1. Divide [a, b] into n equal subintervals, each of width Δx = (b-a)/n 2. For each subinterval, evaluate the function at the left endpoint 3. Sum the areas: S = sum of f(xᵢ) × Δx

As n → infinity

The Riemann sum converges to the exact definite integral. More rectangles = better approximation.

Worked Example

Approximate the integral of x² from 0 to 1 with 10 rectangles.

exp_val = 2a_val = 0b_val = 1num_rect = 10
  1. 01Δx = (1-0)/10 = 0.1
  2. 02Left endpoints: 0, 0.1, 0.2, ..., 0.9
  3. 03Sum = (0² + 0.1² + 0.2² + ... + 0.9²) × 0.1
  4. 04Exact value = 1³/3 = 0.333333...

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