Riemann Sum Calculator

Approximate Definite Integrals

Calculate Riemann sums to approximate definite integrals using left, right, midpoint, or trapezoidal methods. Our calculator helps visualize integral approximations and understand the fundamental theorem of calculus.

Approximate Area Under a Curve

Enter a function, interval, number of subintervals, and select a method to approximate the integral using Riemann sums.

Function, f(x):

Interval Start (a):

Interval End (b):

Number of Subintervals (n):

Method:

Understanding Riemann Sums

Riemann sums are a method of approximating the definite integral of a function over an interval. They work by dividing the area under the curve into smaller rectangles (or trapezoids) and summing their areas.

Methods of Riemann Sums

Left Riemann Sum: Uses the left endpoints of subintervals to calculate the height of rectangles.
Right Riemann Sum: Uses the right endpoints of subintervals for heights.
Midpoint Riemann Sum: Uses the midpoints of subintervals for heights.
Trapezoidal Rule: Approximates the area using trapezoids instead of rectangles.

Formula

The general formula for a Riemann sum is:

S = \sum_{i=1}^{n} f(x_i^*) \Delta x

Where:

$n$ is the number of subintervals.
$\Delta x = \frac{b - a}{n}$ is the width of each subinterval.
$x_i^*$ is a point in the $i$ -th subinterval (left endpoint, right endpoint, or midpoint).

Understanding Riemann Sums

Riemann sums approximate the area under a curve by dividing it into rectangles. Methods: Left sum uses left endpoints, Right sum uses right endpoints, Midpoint sum uses rectangle centers, Trapezoidal uses trapezoids instead of rectangles. More subdivisions (larger n) increase accuracy.

Convergence to Integral

As the number of rectangles approaches infinity (and width approaches zero), the Riemann sum converges to the exact integral value. This is the foundation of integral calculus. The definite integral ∫[a,b] f(x)dx is the limit of Riemann sums as subdivision width → 0.

Choosing the Right Method

Left/Right sums: simple but often less accurate. Midpoint: generally more accurate than left/right. Trapezoidal: good balance of accuracy and simplicity. Simpson's rule (not a Riemann sum): most accurate for smooth functions. Use method appropriate for your accuracy requirements and computational resources.