How does the trapezoidal rule work?

It joins adjacent endpoint samples with straight lines and sums the resulting trapezoid areas.

Can I increase the accuracy?

Use more subintervals and compare the result with another count; it remains an approximation.

Trapezoidal Rule Calculator

Opens the integral calculator in trapezoidal mode. Enter the function, bounds and number of equal strips to calculate and draw the approximation.

Integral0.34375∫ x² dx from 0 to 1 · read as x^2 · approximation, trapezoid rule4 trapezoid strips under the curve between x = 0 and x = 1. They add up to 0.34375.

Integrand

Method

Result: 0.34375
Variable
Bounds
Numeric: 0.34375

Subintervals

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 strips of width 0.25: every interior sample counts once and the two end samples count half each, which is what the trapezoid rule weights them at.

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

How does the trapezoidal rule work?: It joins adjacent endpoint samples with straight lines and sums the resulting trapezoid areas.
Can I increase the accuracy?: Use more subintervals and compare the result with another count; it remains an approximation.

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.