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Riemann Sum Calculator

Compute midpoint Riemann sum approximations with an interactive rectangle visualization and n slider.

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Documentation

What Is a Riemann Sum?

A Riemann sum approximates the area under a curve by dividing the region into rectangles. The interval $[a, b]$ is split into $n$ subintervals of width $\Delta x = \frac{b - a}{n}$ :

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

where $x_i^*$ is the sample point in the $i$ -th subinterval.

Types of Riemann Sums

Type	Sample point	Formula	Tendency
Left	$x_i^* = x_{i-1}$	$\sum_{i=0}^{n-1} f(x_i) \Delta x$	Underestimates (if f increasing)
Right	$x_i^* = x_i$	$\sum_{i=1}^{n} f(x_i) \Delta x$	Overestimates (if f increasing)
Midpoint	$x_i^* = \frac{x_{i-1}+x_i}{2}$	$\sum_{i=1}^{n} f(\bar{x}_i) \Delta x$	More accurate than left or right

Convergence to the Integral

As $n \to \infty$ , all Riemann sums converge to the definite integral (for continuous functions):

\int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \, \Delta x

Error rates: Left and right sums have error $O(1/n)$ . Midpoint sums have error $O(1/n^2)$ — much faster convergence. For even better accuracy, use the trapezoidal rule or Simpson's rule.

Worked Example

Estimate $\int_0^4 x^2 \, dx$ with $n = 4$ right Riemann sum:

\Delta x = 1, \quad x_i = 1, 2, 3, 4

S_4 = [f(1) + f(2) + f(3) + f(4)] \times 1 = [1 + 4 + 9 + 16] = 30

Exact answer: $\frac{x^3}{3}\Big|_0^4 = \frac{64}{3} \approx 21.33$ . The right sum overestimates because $x^2$ is increasing.

Frequently Asked Questions

What is a Riemann sum?

A Riemann sum approximates the area under a curve by dividing the region into rectangles. The interval $[a, b]$ is split into $n$ subintervals, each of width $\Delta x = \frac{b - a}{n}$ , and the function is sampled at a chosen point in each subinterval.

What are the different types of Riemann sums?

The three main types are left, right, and midpoint Riemann sums. Left sums sample at the left endpoint of each subinterval, right sums at the right endpoint, and midpoint sums at the center. Midpoint sums are generally the most accurate.

Do Riemann sums always converge to the exact integral?

Yes, for continuous functions, all Riemann sums converge to the definite integral as the number of rectangles $n$ approaches infinity. Left and right sums converge at rate $O(1/n)$ , while midpoint sums converge faster at $O(1/n^2)$ .

Why does a left Riemann sum underestimate for increasing functions?

For an increasing function, the left endpoint of each subinterval is the smallest value. The rectangle height is therefore less than the actual curve height across most of the subinterval, causing the total area to be underestimated.

How many rectangles do I need for a good approximation?

It depends on the function. For smooth, slowly varying functions, 10 to 50 rectangles may suffice for a rough estimate. For highly curved or oscillatory functions, you may need hundreds or thousands. Simpson's rule typically gives better accuracy with fewer subdivisions.

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Riemann Sum Calculator

Compute midpoint Riemann sum approximations with an interactive rectangle visualization and n slider.

Try an Example

Basic Polynomial

Trigonometric Integral

Area Under a Bell Curve

What Is a Riemann Sum?

Types of Riemann Sums

Convergence to the Integral

Worked Example

Frequently Asked Questions

What is a Riemann sum?

What are the different types of Riemann sums?

Do Riemann sums always converge to the exact integral?

Why does a left Riemann sum underestimate for increasing functions?

How many rectangles do I need for a good approximation?

Related method Variants

Simpson's Rule

Trapezoidal Rule

Midpoint Rule

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