What is a Riemann sum?

A Riemann sum approximates the area under a curve by dividing the interval into n equal subintervals, drawing a rectangle in each subinterval, and summing the rectangle areas. The height of each rectangle depends on which endpoint or midpoint you use to sample the function. As n increases the approximation approaches the definite integral.

What is the difference between left, right, and midpoint Riemann sums?

The left sum uses the function value at the left edge of each subinterval as the rectangle height. The right sum uses the right edge. The midpoint sum uses the function value at the centre of each subinterval, which is generally the most accurate of the three for the same number of rectangles because errors on either side partially cancel.

How many rectangles should I use?

More rectangles give a more accurate approximation. For a smooth function like x squared, 6 rectangles gives a rough estimate and you can see the left sum undershooting and the right sum overshooting. Use 100 or more rectangles to get an approximation close to the true integral value. The midpoint rule converges to the true value roughly four times faster than the left or right rules.

Riemann Sum Calculator

A Riemann sum approximates the definite integral of a function by dividing the region under the curve into a finite number of rectangles and summing their areas. It is the foundation of integral calculus and gives you a concrete way to see how the exact integral emerges as the number of rectangles grows without limit. The three most common versions differ only in where on each subinterval you sample the function to set the rectangle height. The left Riemann sum samples the left endpoint of each subinterval, the right sum samples the right endpoint, and the midpoint sum samples the centre. For an increasing function the left sum will underestimate and the right sum will overestimate the true area, with the midpoint sum generally splitting the error and converging fastest. This calculator accepts any function of x in standard notation, the lower limit a and upper limit b of integration, and the number of rectangles n. It returns the left sum, the right sum, and the midpoint sum simultaneously so you can see how the three approximations compare. You can use standard operators +, -, *, / and ^ for powers, along with functions such as sin, cos, exp, log, sqrt, and abs. Increase n to watch all three sums converge toward the true integral. The default example uses f(x) = x^2 from 0 to 3 with 6 rectangles, which gives a clear demonstration of the difference between the three methods.

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Function f(x)

Lower limit a

Upper limit b

Number of rectangles n

8.938

midpoint Riemann sum

Left sum6.875

Right sum11.375

True integral (Simpson's)9.000

Exact integral uses Simpson's rule with 1000 subintervals for reference. Use ^ for powers and * for multiplication.

How it works

The interval [a, b] is divided into n equal subintervals each of width dx = (b - a) / n. For the left sum, the height of each rectangle is f evaluated at the left edge of the subinterval: f(a + i * dx) for i from 0 to n-1. For the right sum, the height is f(a + (i+1) * dx) for i from 0 to n-1. For the midpoint sum, the height is f at the centre: f(a + (i + 0.5) * dx). Each sum multiplies the rectangle height by dx and accumulates the total. The exact reference uses Simpson's rule with 1000 subintervals, which is accurate to many decimal places for smooth functions.

Worked example

For f(x) = x^2 from a = 0 to b = 3 with n = 6 rectangles, each rectangle has width dx = 0.5. The left endpoints are 0, 0.5, 1, 1.5, 2, 2.5 and the left sum is (0 + 0.25 + 1 + 2.25 + 4 + 6.25) * 0.5 = 6.875. The right endpoints are 0.5, 1, 1.5, 2, 2.5, 3 and the right sum is (0.25 + 1 + 2.25 + 4 + 6.25 + 9) * 0.5 = 11.375. The midpoints are 0.25, 0.75, 1.25, 1.75, 2.25, 2.75: f values are 0.0625, 0.5625, 1.5625, 3.0625, 5.0625, 7.5625. Sum * 0.5 = (0.0625 + 0.5625 + 1.5625 + 3.0625 + 5.0625 + 7.5625) * 0.5 = 17.875 * 0.5 = 8.938. The true integral of x^2 from 0 to 3 is exactly 9, shown by Simpson's rule reference.

Related calculators

Area Under the Curve Calculator: compute the exact area using Simpson's rule integration.
Average Rate of Change Calculator: find the slope of a secant line over an interval.
Tangent Line Calculator: find the tangent to a curve at a single point.
Series Sum Calculator: sum arithmetic and geometric series.

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