Is a Riemann sum exact?

No. It is a numerical approximation whose accuracy depends on the function, interval and number of subintervals.

Which rectangle rule is available?

Choose left, right or midpoint heights; the page labels the selected rule and draws the strips.

Riemann Sum Calculator

Opens the integral calculator in Riemann mode. Enter an integrand, bounds and strip count to compare left, right and midpoint rectangle estimates.

Integral0.21875∫ x² dx from 0 to 1 · read as x^2 · approximation, rectangle rule4 rectangles under the curve between x = 0 and x = 1, height taken at the left. They add up to 0.21875.

Integrand

Method

Result: 0.21875
Variable
Bounds
Numeric: 0.21875

Subintervals

Height taken at

The numerical methods add up strips of one variable between two numbers, so they take a single variable and a pair of bounds rather than an iterated integral.

4 rectangles of width 0.25, each one as tall as the integrand at the left of its subinterval.

How far this sits from the true integral is not known from the sum alone: an error bound needs a limit on the integrand's curvature, which this page does not work out. Raise the count and watch the figure settle.

x^2 dxindefinite
x³/3 + C
x^2 from 0 to 3definite
9
exp(x^2) dxno elementary form
named, not approximated

Accuracy. Exact symbolic integration for the functions it supports, with the expression it actually read printed back so you can check the parse. An integral with no elementary antiderivative is named as such rather than approximated silently, and no step-by-step derivation is claimed.

What you can type

Numbers, single letters for variables, the constants pi and e, the signs + - * / ^ and brackets, and these functions: sin, cos, tan, sec, csc, cot, asin, acos, atan, sinh, cosh, tanh, exp, sqrt, abs, log. ln is read as log, which is the natural logarithm here, and arcsin, arccos and arctan are read as asin, acos and atan. Multiplication can be left out where it is obvious, so 2x, 3 sin(x) and 2(x + 1) all read the way you would write them on paper, and the stage prints back what that came to. Leave both bounds empty for an antiderivative, or fill them in for a definite integral. Powers go up to 64, and the expression can be up to 500 characters long.

Common questions

Is a Riemann sum exact?: No. It is a numerical approximation whose accuracy depends on the function, interval and number of subintervals.
Which rectangle rule is available?: Choose left, right or midpoint heights; the page labels the selected rule and draws the strips.

Exact symbolic integration for the functions it supports, with the expression it actually read printed back so you can check the parse. An integral with no elementary antiderivative is named as such rather than approximated silently, and no step-by-step derivation is claimed.