Riemann Sum Visualizer

Enter a function and interval, choose a Riemann method, and drag n to watch the rectangles converge to the exact definite integral.

rectangles overestimate underestimate f(x)

approximation

3.19

exact

2.67

error

0.52

error vs. n

General

R_{n} = \sum_{i = 1}^{n} f (x_{i}^{*}) Δ x

Step size

Δ x = \frac{b - a}{n} = \frac{2 - 0}{8} = 0.25

Sample pt

x_{i}^{*} = a + i Δ x = 0.25 i

Substituted

R_{8} = \sum_{i = 1}^{8} [(0.25 i)^{2}] \cdot 0.25

Expanded

= [(0.25)^{2} + (0.5)^{2} + \dots + (2)^{2}] \cdot 0.25

Numerical

R_{8} \approx 3.19

Integral

\int_{0}^{2} x^{2} d x = n \to \infty lim R_{n} = 2.67